7/4/23, 11:16 PM Homework 8.7-Anh Nguyen Use a table of integrals to determine tv7t+ 19 at. Click here to view page 1 of the integral table. " Click here to view page 2 of the integral table. 14 Click here to view page 3 of the integral table. " Click here to view page 4 of the integral table. 16 Click here to view page 5 of the integral table." Click here to view page 6 of the integral table. 18 JWV7t + 19 at= 29 3(71+ 19)2 3 ( 7 1 + 19 ) 2 + c 13: Table of Integrals (page 1) Substitution Rule g(b) ( 19(x)g' (x) ax = fru ) au (u =9(x ) Srg(xng'(x ) ax = ru ) du g(a) Integration by Parts Judv=uv - frau fur' ax =uvid - Jvu ax Basic Integrals J X" dx = =17x +1 + C;n# -1 * = In / x/ + c cos ax dx = = sin a sin ax dx = - - cos ax + C tan x dx = In |secx | + C cot x dx = In | sin x| secx dx = In |sec x + tan x| + C csc x dx = - In cscx + cotx| + C eax dx = =eax + c bax dx = = a In b bax + C; b > 0, b#1 In x dx = xInx - x + C log , X dx = In b (XIl dx = = sin Va2 - + C dx dx 2 - = - tan - 12 + c . sec a a x1x2 - a a sin - 'x dx = xsin -1 x + $1 - x2 + c cos " 'x dx = x cos - 1x - 1 1 - x2 + c tan - 'x dx = x tan sec 'xax=xsec -1x - In (x + x2 - 1 ) + c sinh x dx = cosh x + C cosh x dx = sinh x . sech 2 x dx = tanh x + C csch 2x dx = - coth x + C sech x tanh x dx = csch x coth x dx= - csch x + C tanh x dx = In coshx + C coth x dx = In | sink sech x dx = tan 'sinh x + C= sin -1 tanh x + C csch x dx = In tann 2 + c 14: Table of Integrals (page 2) Trigonometric Integrals https://xlitemprod.pearsoncmg.com/api/v1/print/highered 10/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen cos 2 x dx = 7 + - sin 2x 4 + C sin 2x dx = 2 sin 2x 4 + C sec 2ax dx = 1 csc 2ax dx = - ~ cotax + C tan 2 x dx = tan x - x + C [ cot 2 x dx = - c cos ' x dx = - = sin 'x + sin x + C sin 'x dx = = cos x- cosx + C sec 3 x dx = 7: csc 3x dx = - - csexcotx- - In /csex+ cotx/ +C tan 3 x dx = = tan 2x - In /secx| +c cot 3x dx = - = sec " ax tan ax dx = - sec "ax + C;n# 0 csc " ax cotax dx = - - csc " ax + C ; n#0 1x 1 + sin ax dx 1 - sin ax - tan - ax + c dx ax 1 + cos ax - tan + C 1 - cos ax cos (m + n)x cos (m - n)x sin mx cos nx dx = 2(m + n) - + C;m #n2 2(m - n) sin (m - n)x sin (m + n)x sin mx sinnx dx = 2(m - n) + C; m2 # n2 2(m + n) cosmx cos nx dx = sin (m - n)x sin (m + n)x 2(m - n) 2(m + n) -+ C ;m # n2 15: Table of Integrals (page 3) Reduction Formulas for Trigonometric Functions cos "xax = " cos " 'xsinx+ " " cos "-2x dx sin "x dx = - - sin "- 1x cos x+ "- tan "'x dx = tan "-1 n - 1 J tan " -2x dx; n#1 cot " x dx = _ cot " -1x n - 1 -[ corn - 2, sec " x dx = s sec " - 2xtan x n- 2 sec " - 2 x dx; n # 1 csc "x dx = _ csc "-2x cotx n + 2 n- 1 n - 1 n- 1 n - 1 sin " x cos "x dx =- sin -1xcos +1x m - 1 m + n m +n.J sin - 2 xcos " x dx ; m # - n sin "xcos "x dx = sin m+1x xcos" -'x n- 1 m + n m +nJ sin "xcos "-2x dx; m # -n x" sin ax dx = _ x x" cos ax x " - cos ax dx ; a # 0 [x" cos ax dx = - x" sin ax a a Integrals Involving a2 - x2; a > o https://xlitemprod.pearsoncmg.com/api/v1/print/highered 11/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen [ Vaz - x2 ax=>Va2-x2 + sin-12+ dx - In a + Va2 -x2 a + C xval -> X + C dx Va2 - x2 x2 Na2 - x2 + C [x2 Ja2 - x2 dx = = (2x2-a?) a2 -x2 + 3 sin-12 +c Va2 -x2 dx = - - va2 - x2 + C- sin-1_ + c x2 X a 2 dx= - 2 Va2 -x2 + 3 sin-1X + dx - In 2a 16: Table of Integrals (page 4) Integrals Involving x2 - a2; a > 0 [ Vx2 - 2 2 ox = * Vx2- a2 - 2- In/ x + 1x2 - a?| +c JJ2 2 = In x +1x - 2 2 | + c dx Vx 2 - 22 + C x vx -a a x [ x2 1x2 - 27 ax= = ( 2x2 - 2 2 ) 1x2- 22 - In/ x+ 1x2- 2?|+c Vx2 - 2 2 dx = In x + 1x2 - a21. Vx2 - a2 + C X Vx 2 - a2 2 dx = 2 In |x + Vx2 -a?| + x x2 -a2 + c dx 2 - 2 2 = - In X - a + c dx 1 In x2 -a2 x ( x 2 - 2 2 ) 2a * 2 + C Integrals Involving a2 + x2; a> 0 ( Vaz + x 2 Ox = = Vaz + x7 + 3- In(x+122 +x] ) +c dx Va 2 + x = = In (x + Va2 +x2 ) +c dx - In a - Val +x2 x dx Na 2 + x 2 xva2 + , + C 1 2 2 2 +x 2 + C a x [ x2 a2 +x2 ax= * (22+ 2x2 ) a2 +x2- 2 in (x+ Va2 +x? ) + c ( Na2 +x2 ax = In /x + a2 +x2| va2 +x2 + C X x2 dx = - 2 In (x + Vaz+x2) + xva2 +x2 a2 + x2 2 .+ C * *- dx = Val + x 2 - aln a + Vaz +x2 + C dx X (a2 + x2) 3/2 2 2 2 2 + x 2 + C dx x ( 2 2 + x 2) 2a2 In + C 17: Table of Integrals (page 5) Integrals Involving ax b; a #0, b > 0 https://xlitemprod.pearsoncmg.com/api/v1/print/highered 12/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen (ax + b)" dx = (ax + b) n + 1 a(n+ 1) - + C; n# - 1 ( Vax + b ) " dx = 2 ( vax + b ) n+ 2 n +2 - + C; n# -2 dx 2 -1 ax - b dx tan - + C Vax + b - vb xvax - b Vb x vax + b = In Vb Vax + b + vb + C X - dx = - - In lax + b| + c ax + b a dx = - ax + b 273 ( (ax +b)2 - 4b(ax + b) +2b2 in lax + bl) + c dx a -+ - ax + b In x (ax + b) + C xVax + b dx = 2 bx x (3ax - 2b)(ax + b)3/2 + c 15a X 2 dx = - b 32 2 (ax - 2b) vax + b + C [ ax + b ax + b | x ( ax + b ) " dx = ( ax + b ) " + 1 2 n + 2 n + 1 + C; n# -1, -2 dx - In X x (ax + b ) ax + b +C Integrals with Exponential and Trigonometric Functions eax sin bx dx = e (a sin bx - bcos bx) + C eax cos bx dx =- e(a cos bx + b sin bx) a + 62 + C a + 62 18: Table of Integrals (page 6) Integrals with Exponential and Logarithmic Functions dx xinx = In linx| + c [x" Inxox= n+1 Inx - n+1 + C;n# - 1 [xe* dx= xex - ex + c Ix" ax ax = 1x"eax _ " fx - lax dx; a #0 In "x dx =xin "x-n In"-1x dx Miscellaneous Integrals x" cos 'xax= 1 x+ 1 cos-1x+ [x *n+ 1 dx Fin# -1 [ x" sin - 'x ax = n+7 xnt 'sin -1x - JX ; n* - 1 x" tan - 1x dx = -xn+ 1 dx n+ 7 x0 + 1 tan - 1 x - -2 4 1 in * - 1 2ax -x2 dx = x-azax - x2 + 3 sin-1 / X-2 +c;a>o dx = sin -1 X - a 2ax - x2 https://xlitemprod.pearsoncmg.com/api/v1/print/highered 13/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen dx Use a table of integrals to determine 81 - 49x2 Click here to view page 1 of the integral table. " Click here to view page 2 of the integral table. 20 Click here to view page 3 of the integral table." Click here to view page 4 of the integral table. 22 Click here to view page 5 of the integral table.Click here to view page 6 of the integral table. 24 dx 81 - 49x2 726 In gx + 1 - In 5 x - 1 + c 19: Table of Integrals (page 1) Substitution Rule g(b) | +(9(x) )9'(x) ax = [ru) du (u= 9(x) [=(9(x)g' (x ) ax = [ fu ) du g(a) Integration by Parts Juav=uv-Svau fuv' ax= uvla - [vu' ax a Basic Integrals | x" dx = 1x+1 + C;n# -1 * = In ( x / + c cos ax dx = = sin a sin ax dx = - - cos ax + C tan x dx = In |secx| + C cot x dx = In |sin x| secx dx = In |sec x + tan x| + C csc x dx = - In |cscx + cotx| + C = Leax + C bax dx = - a In b _ bax + C; b > 0, b#1 In x dx = x Inx - x + C log b x dx = 1 In b (Xli dx = = sin - 12 + 5 + C dx dx Va2 2 - 2 = - tan - 12 + c - sec a a x1/x 2_ a sin - 'xdx=xsin -1x +1-x2 +c cos - 'x dx = x cos - 1x - 1-x2 + c tan - 1x dx = x tan sec 'xax=xsec -'x - In (x + Vx2 - 1) +c sinh x dx = cosh x + C cosh x dx = sinh x sech 2 x dx = tanh x + C csch 2x dx = - coth x + C sech x tanh x dx = csch x coth x dx= - csch x + C tanh x dx = In coshx + C coth x dx = In | sink sech x dx = tan 'sinh x + C = sin -1 tanh x + C csch x dx = In tann 2 + c 20: Table of Integrals (page 2) Trigonometric Integrals https://xlitemprod.pearsoncmg.com/api/v1/print/highered 14/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen cos 2 x dx = 7 + - sin 2x 4 + C sin 2x dx = 2 sin 2x 4 + C sec 2ax dx = 1 csc 2ax dx = - ~ cotax + C tan 2 x dx = tan x - x + C [ cot 2 x dx = - c cos ' x dx = - = sin 'x + sin x + C sin 'x dx = = cos x- cos x + C sec 3 x dx = 7: csc 3x dx = - - csexcotx- - In csex+ cotx/ +C tan 3 x dx = - tan 2x - In /secx| + c cot 3x dx = - = sec " ax tan ax dx = = - sec "ax + C ;n # 0 csc " ax cot ax dx = - csc "ax + C ;n # 0 1x 1 + sin ax dx - tan - ax + c dx ax 1 - sin ax 1 + cos ax - tan ? + C 1x 1 - cos ax cos (m + n)x cos (m - n)x sin mx cos nx dx = 2(m + n) - + C;m # n2 2(m - n) sin (m - n)x sin (m + n)x sin mx sinnx dx = 2(m - n) 2(m + n) + C; m2 # n2 cosmx cos nx dx = sin (m - n)x sin (m + n)x 2(m - n) 2(m + n) -+ C ;m # n2 21: Table of Integrals (page 3) Reduction Formulas for Trigonometric Functions cos "xax = " cos" 'xsinx+ " " cos "-2x dx sin "x dx = - - sin "- 1x cos x+ "- tan "'x dx = tan nn-1 x n - 1 J tan "-2 x dx; n#1 cot " x dx = _ cot " -1x n - 1 -[ corn - 2, sec "x dy = sec " -2x tan x n- 2 n - 1 sec " - 2 x dx; n # 1 csc "x dx = _ csc"-2x cotx n * 2 n- 1 n- 1 n - 1 sin " x cos "x dx = - sin m -1x cos +1x m - 1 m + n m +nJ sin m- 2 xcos " x dx ; m # - n sin "xcos "x dx = sin m+1x xcos "-1x n- 1 m + n m +nJ sin "xcos "-2x dx; m # -n x" sin ax dx = _ x x" cos ax x " - cos ax dx ; a # 0 [x" cos ax dx = - x" sin ax a a Integrals Involving a2 - x2; a > o https://xlitemprod.pearsoncmg.com/api/v1/print/highered 15/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen 5. Use a table of integrals to evaluate the indefinite integral. This integral requires preliminary work, such as completing the square or changing variables, before it can be found in a table. 2ex =dx 149 + 4e2X 25 Click here to view a table of integrals Which of the following is the value of the given integral? O A. In 149 + 4e2X + c OB. In (u + 149 + u2 ) + c O c. In (ex + 149+ e2x ) + c &D. In (2e* + 49+4e2X ) + c 25: Reference Table of Integrals dx 2 = tan -1 ax - b + C x vax - b Vb b [ Vaz + x2 ax = " Va2 + x2 + In/x+Va2+x2|+c dx 13 2 + x 2 = In /x + Va2 +x2 + c dx 1 + sin ax tan 4 - 2 dx 2 2ax + b - tan ax + bx + c V 4ac - b 14ac - b2 a-x dx= - (x(a -x) -atan Vx (a - x) x - a https://xlitemprod.pearsoncmg.com/api/v1/print/highered 18/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen . Use a table of integrals to evaluate the following indefinite integral. The integrals may require preliminary work, such as completing the square or changing variables, before it can be found in a table. VIn Ex + 25 -dx Click here to view page 1 of the integral table. Click here to view page 2 of the integral table. 27 Click here to view page 3 of the integral table. 2 Click here to view page 4 of the integral table. 29 Click here to view page 5 of the integral table. Click here to view page 6 of the integral table. 31 In 2x + 25 "I - ax = (125 + In2(x) In (x)+25 In (Use parentheses to clearly denote the argument of each function.) 26: Table of Integrals (page 1) Substitution Rule g (b ) "(9(x)9' (x ) ax = [ru ) du ( u = g ( * ) ) J (9(x)g'(x) ax= =(u ) du a g(a) Integration by Parts Juav=ur-Jvau juvax= wv/2- fvwax Basic Integrals [ x" dx = 4 xxn+ 1 + C;n* - 1 = In / x / + c cos ax dx = = sin ax + C sin ax dx = - - cosax + C tan x dx = In /sec x | + C cotx dx = In | sin x | + c secx dx = In |sec x + tan x| + C csc x dx = - In /cscx + cotx/ + C Jeax ax = - eax + C bax dx = = alnb - bax + C; b > 0, b#1 In x dx = xIn x - x + C 109 , * dx = 1 In b ( X In x - x ) + C dx = sin - 12 + c dx Va2 - x 2 = - tan - 12 + c dx - sec -1 +c a x7/ x2 - a2 a sin - 'x dx = xsin - 1 x + $1- x2 + c cos -'x dx=xcos -1x- $1-x2 + c tan - 'x dx =xtan - 1x-> In( sec - 1x dx =xsec- 1x - In (x + (x2 - 1 ) + c sinh x dx = cosh x + C cosh x dx = sinh x + C sech 2 x dx = tanhx + C csch 2 x dx = - coth x + C sech x tanh x dx = - sechx + ( csch x coth x dx = - cschx + C tanh x dx = In coshx + c coth x dx = In | sinh x | + C sech x dx = tan - 1 sinh x + C= sin -1 tanh x + C cschx dx = In tann ~ + c 27: Table of Integrals (page 2) Trigonometric Integrals https://xlitemprod.pearsoncmg.com/api/v1/print/highered 19/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen cos 2 x dx = 2* sin 2x 4 +C sin 2x dx = sin 2x + C 4 sec 2 ax dx = ~ tan ax + C csc 2 ax dx = - - cotax + C tan 2 * dx = tan x - x + C cot 2 x dx = - cotx - x + C |cos 3 x dx = - 7 sin 3x + sinx + C sin 3 x dx = = cos 3x - cos x + C sec 3 x dx = = sec x tanx - csc 3 x dx= - > csexcotx - > In /cscx+ cotx| + C tan 3 x dx =- tan 2x - In /secx| + c cot 3 x dx = - cot 2x - 11 sec " ax tan ax dx = = sec "ax + C; n#0 csc " ax cotax dx = - macscax + C; n#0 dx 1 + sin ax - tan dx 1tan II , ax + c dx ax + C - tan dx a ax 1 - sin ax 1 + cos ax a 1 - cos ax a 2 sin mx cos nx dx = - cos (m + n)x cos (m - n)x - + C; m2 #n2 2(m + n) 2(m - n) sin mx sin nx dx = sin (m - n)x sin (m + n)x 2(m - n) 2(m + n) + C; m2 # n2 cos mx cos nx dx = sin (m - n)x sin (m + n)x 2(m - n) 2(m + n) + C; m2 # n2 28: Table of Integrals (page 3) Reduction Formulas for Trigonometric Functions cos " xax = - cos"-'xsin x + "-1 / cos "-2x dx sin "xax = - 1 sin "- 'x cos x + tan "'x dx = tan " -1x - J tan "-2x dx; n # 1 cot "'x dx = _ cot " -1x n - 1 n - 1 cot 17 -+2, sec " x dx = - _ sec " -2xtanx n-2 csc "x dx= _ cc "-2xcotx n-2 n- 1 n - 7 / sec " - 2 x dx ; n # 1 n - 1 n - 1 sin "xcos "x dx= _ sin"-'xcos "+1x m - 1 m + n m + n . sin m- 2 xcos "x dx ; m # - n sin "xcos "x dy sin " 'xcos "-1x n - 1 m + n m +n sin "xcos " - 2 x dx ; m# -n x" sin ax dx = _ X" cos ax a 2 x" - 1 cos ax dx; a * 0 [ x" cos ax dx = X sin ax a Integrals Involving a2 -x2; a > 0 https://xlitemprod.pearsoncmg.com/api/v1/print/highered 20/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen [ Vaz - x2 ax=>Va2-x2 + sin-12+ dx a + C - In a + Va2 -x2 xval -> X + C dx Va2 - x2 x2 Na2 - x2 + C [x2 Ja2 - x2 dx = = (2x2-a?) a2 -x2 + 3 sin-12 +c Va2 -x2 dx = - - va2 - x2 + C- sin-1_ + c x2 X a 2 dx= - 2 Va2 -x2 + 3 sin-1X + dx - In 2a 29: Table of Integrals (page 4) Integrals Involving x2 - a2; a > 0 [ Vx2 - 2 2 ax = * Vx2- 22 - 3- In / x + 1x2 - a? | + c JJ2 2 = In x +1x - 2 2 | + c dx Vx 2 - 22 + C x vx -a a x [ x2 1x2 - 27 ax= = ( 2x2 - 2 2 ) 1x2- 22 - In/ x+ 1x2-2?|+c Vx2 - 2 2 dx = In x + 1x2 - a21. Vx2 - a2 + C X Vx 2 - a2 2 dx = 2 In |x + Vx2 -a?| + x x2 -a2 + c dx 2 - 2 2 = - In X - a + c dx x ( x 2 - 2 2 ) 1 In x2 -a2 2a * 2 + C Integrals Involving a2 + x2; a> o ( Vaz + x 2 Ox = = Vaz + x7 + 3- In(x+122 +x] ) +c dx Va 2 + x = = In (x + Va2 +x2 ) + c dx - In a - Val +x2 x + C dx Na 2 + x 2 xva2 + , 1 2 2 2 +x 2 + C a x [ x2 Vaz +xox= x (a2+ 2x2) a2 +x2- 2 in (x+1a2 +x] ) + c (Na2 +x2 ax = In |x + Va2 + x2| va2+x2 + C X x2 2 2 + x 2 dx = - 2 In (x + Va2 +x2) + xva2+x2 2 + C [Na? +x2 * *- dx = Na? + x 2 - aln a+ Vaz+x2 + C dx X (a2 +x2) 3/2 + C a2 Va2 + x2 dx * ( 2 2 + x 2 ) - In 2a2 a2 + x2 + C 30: Table of Integrals (page 5) Integrals Involving ax = b; a #0, b > 0 https://xlitemprod.pearsoncmg.com/api/v1/print/highered 21/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen (ax + b)" dx = (ax + b) n + 1 a(n+ 1) - + C; n# - 1 ( Vax + b ) " dx = 2 ( vax + b ) n+ 2 n +2 - + C; n# -2 dx 2 dx tan -1 ax - b Vax + b - vb xvax - b Vb - + C x vax + b = In Vb Vax + b + vb + C X - dx = - - In lax + b| + c ax + b a dx = - ax + b 273 ( (ax +b)2 - 4b(ax + b) +2b2 in lax + bl) + c dx a -+ - ax + b In x (ax + b) + C xVax + b dx = 2 bx x (3ax - 2b)(ax + b)3/2 + c 15a X 2 dx = - 2 (ax - 2b) vax + b + C | x ( ax + b ) " dx = ( ax + b ) " + 1 [ ax + b ax + b b 32 2 n + 2 n + 1 + C; n# -1, -2 dx x (ax + b ) - In X ax + b +C Integrals with Exponential and Trigonometric Functions eax sin bx dx = e (a sin bx - bcos bx) + C e(a cos bx + b sin bx) a + 62 eax cos bx dx =- + C a + 62 31: Table of Integrals (page 6) Integrals with Exponential and Logarithmic Functions dx xin x = In |In x| + c [x" Inxox= n+1 Inx - n+1 + C;n# - 1 [xe* dx= xex - ex + c Ix ax ax = 1x"eax _ " fx - lax dx; a #0 In "x dx =xin "x-n In"-1x dx Miscellaneous Integrals x" cos 'xax= 1 x+ 1 cos-1x+ [x *n+ 1 dx Fin# -1 [ x" sin - 'x ax = n+7 xnt 'sin -1x - JX ; n* - 1 x" tan - 1x dx = -xn+ 1 dx n+ 7 x0 + 1 tan - 1 x - -2 4 1 in * - 1 2ax -x2 dx = x-azax - x2 + 3 sin-1 / X-2 +c;a>o dx = sin -1 X - a 2ax - x2 https://xlitemprod.pearsoncmg.com/api/v1/print/highered 22/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen 7. Use a table of integrals to evaluate the following indefinite integral. This integral requires preliminary work, such as completing the square or changing variables, before it can be found in a table cos X -dx 6 sin *x + 5 sin x Click here to view page 1 of the integral table.$2 Click here to view page 2 of the integral table. 33 Click here to view page 3 of the integral table. 34 Click here to view page 4 of the integral table. 35 Click here to view page 5 of the integral table. 36 Click here to view page 6 of the integral table. 37 cos x dx = 5 In |sin (x)| - 5 In 16 sin (x) +5| + c 6 sin *x + 5 sin x 32: Table of Integrals (page 1) Substitution Rule g (b) J (9(x/19'(x) ax = [i(u ) du ( u = g( x ) ) |+(9(x)g'(x) ox= [ fu ) du a g(a) Integration by Parts Juav=uv-fvdu Juvax = uv/2 - Svu'dx Basic Integrals x " dx = 1 n + 7 X" + 1 + C;n# -1 J = In / x / + c cosax dx = = sin ax + C sin ax dx = - - cosax + C tan x dx = In /sec x | + C cotx dx = In | sin x | + c secx dx = In / sec x + tan x/ + C csc x dx = - In /cscx + cotx| + C Jeax ax = -eax + c bax dx = - 1 alnbbax + c; b > 0, b #1 In x dx = * In x - * + C |109 , x dx = in/ ( x Inx - x ) + C dx = sin - 14 + c dx = - tan - 12 + c dx x 1 x2 - a2 = -sec-1 * +c a a sin - 'x dx = x sin - 1 x + 1 1- x2 + c cos - 1xdx=xcos- 1x- 1-x2 + c tan 'x dx =xtan - 1x - In ( sec - 1x dx =xsec -1x - In (x + x2 - 1 ) +c sinh x dx = cosh x + C cosh x dx = sinh x + C sech 2 x dx = tanhx + C csch 2x dx = - coth x + C sech xtanh x dx = - sechx + ( csch x coth x dx = - cschx + C tanh x dx = In coshx + C coth x dx = In | sinh x| + c sech x dx = tan 'sinh x + C= sin -1 tanh x + C cschx dx = In tanh * + c 33: Table of Integrals (page 2) Trigonometric Integrals https://xlitemprod.pearsoncmg.com/api/v1/print/highered 23/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen cos 2 x dx = 2* sin 2x 4 +C sin 2x dx = sin 2x + C 4 sec 2 ax dx = ~ tan ax + C csc 2 ax dx = - - cotax + C tan 2 * dx = tan x - x + C cot 2 x dx = - cotx - x + C |cos 3 x dx = - = sin 3x + sinx + C sin 3 x dx = = cos 3x - cos x + C sec 3 x dx = = sec x tanx - csc 3 x dx= - > csexcotx - > In /cscx+ cotx| + C tan 3 x dx = - tan 2x - In /secx| + c cot 3 x dx = - cot 2x - 11 sec " ax tan ax dx = = sec "ax + C; n#0 csc " ax cotax dx = - macscax + C; n#0 dx 1 + sin ax - tan dx dx 1 - sin ax a 1tan II , ax + c ax + C 1 + cos ax - tan dx ax a 1 - cos ax a 2 sin mx cos nx dx = - cos (m + n)x cos (m - n)x - + C; m2 #n2 2(m + n) 2(m - n) sin mx sin nx dx = sin (m - n)x sin (m + n)x 2(m - n) 2(m + n) + C; m2 # n2 cos mx cos nx dx = sin (m - n)x sin (m + n)x 2(m - n) 2(m + n) + C; m2 # n2 34: Table of Integrals (page 3) Reduction Formulas for Trigonometric Functions cos "xax = = cos"- 'xsin x+ "-! cos -2x dx sin "x dx = - - sin "- 'x cos x+ | tan "'x dy = tan " -1x n - 1 tan "-2 x dx; n# 1 |cot " x dx = _ cot " -1x n - 1 [ cot 1 - 2, |sec "x dx = sec " 2xtan x n- 2 n- 1 n-1 ) sec " -2 x dx; n#1 csc x dx= _ csc "-2x cotx n-2 n - 1 n - 1 sin "xcos "x dx= _ sin" 'xcos "+1x m - 1 m + n m + nJ sin m-2 xcos "x dx; m # - n sin "xcos "x dy sin "* 'xcos "-1x n - 1 m + n m +n sin " xcos " - 2 x dx ; m # -n x" sin ax dx = _ X" cos ax a 2 x" - 1 cos ax dx; a * 0 [ x" cos ax dx = X sin ax a Integrals Involving a2 -x2; a > 0 https://xlitemprod.pearsoncmg.com/api/v1/print/highered 24/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen [ a2 - x2 ax=>Va2 -x2 + sin-12+ dx a + C - In a + Va2 -x2 xval -> X + C dx Va2 - x2 x2 Na2 -x2 + C [x2 Ja2 - x2 dx = = (2x2-a?) a2 -x2 + 3 sin-12 + c Va2 -x2 dx = - - va2 - x2 + C- sin-1_ + c x2 X a 2 dx= - 2 Va2 - x2 + 3 sin-1X + dx - In 2a 35: Table of Integrals (page 4) Integrals Involving x2 - a2; a> 0 [ Vx2 - 2 2 ax = * Vx2- 22 - 3- In / x + 1x2 - a? | + c JJ2 2 = In x +1x - 2 2 | + c dx Vx 2 - 22 + C x vx -a a x [ x2 1x2 - 27 ax= = ( 2x2 - 2 2 ) 1x2- 22 - In/ x+ 1x2- 2?|+c Vx2 - 2 2 dx = In x + 1x2 - a2| Vx2 - a2 + C X Vx 2 - a2 2 dx = 2 In |x + Vx2-a?| + x x2 -a2 + c dx 2 - 2 2 = - In X - a + c dx x ( x 2 - 2 2 ) 1 In x2 -a2 2a * 2 + C Integrals Involving a2 + x2; a> o ( Vaz + x 2 Ox = = Vaz + x7 + 3- In(x+122 +x] ) +c dx Va 2 + x = = In (x + Va2 +x2 ) + c dx - In a - Val +x2 x + C dx Va 2 + x 2 xva2 + , 1 2 22 +x 2 + C a x [ x2 Vaz +xox= x (a2+ 2x2 ) a2 +x2- 2 in (x+1a2 +x] ) + c ax = In |x + Va2 + x2| va2+x2 + C X x2 2 2 + x 2 dx = - 2 In (x + Va2 +x2) + xva2+x2 2 + C [Na? +x2 * *- dx = Na? + x 2 - aln a+ Vaz+x2 + C dx X (a2 +x2) 3/2 + C a2 Va2 + x2 dx * ( 2 2 + x 2 ) - In 2a2 a2 + x2 + C 36: Table of Integrals (page 5) Integrals Involving ax = b; a#0, b > 0 https://xlitemprod.pearsoncmg.com/api/v1/print/highered 25/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen (ax + b)" dx = (ax + b) n + 1 a(n+ 1) - + C; n# - 1 ( Vax + b ) " dx = 2 ( vax + b ) n+ 2 n +2 - + C; n# -2 dx 2 -1 ax - b dx Vax + b - vb xvax - b tan Vb - + C x vax + b = In Vb Vax + b + vb + C X - dx = - - In lax + b| + c ax + b a dx = - ax + b 273 ( (ax + b)2 - 4b(ax + b) +2b2 in lax + bl) +c dx a -+ - ax + b In x (ax + b) x + C xVax + b dx = 2 bx (3ax - 2b)(ax + b)3/2 + c 15a X 2 dx = - 2 (ax - 2b) vax + b + C [ ax + b ax + b b 32 | x ( ax + b ) " dx = ( ax + b ) " + 1 2 n + 2 n + 1 + C; n# -1, -2 dx x (ax + b ) - In X ax + b +C Integrals with Exponential and Trigonometric Functions eax sin bx dx = e (a sin bx - bcos bx) + C e(a cos bx + b sin bx) a + 62 eax cos bx dx =- + C a + 62 37: Table of Integrals (page 6) Integrals with Exponential and Logarithmic Functions dx xinx = In |In x| + c [x" Inxox= n+1 Inx - n+1 + C;n# - 1 [xe* dx= xex - ex + c Ix ax ax = 1x"eax _ " fxn - lax dx; a #0 In "x dx =xin "x-n In"-1x dx Miscellaneous Integrals x" cos -1x x = 4 x+ 1 cos- 1x+ [x *n+ 1 dx Fin# -1 [ x" sin - 1xox = n+7 xnt 'sin - 1x - J ; n* -1 x" tan - 1x dx = -xn+ 1 dx 1+ 7 X0* 1 tan - 1 x - J -2 4 1 in * - 1 2ax -x2 dx = x-azax - x2 + 3 sin-1 / X-2 +c;a>o dx = sin -1 X - a 2ax - x2 https://xlitemprod.pearsoncmg.com/api/v1/print/highered 26/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen 8. Use integral tables to evaluate the integral. 1 - dx x1/49 -x2 Click here to view page 1 of the Table of Integrals. Click here to view page 2 of the Table of Integrals. 39 Click here to view page 3 of the Table of Integrals. 40 Click here to view page 4 of the Table of Integrals. 41 Click here to view page 5 of the Table of Integrals."Click here to view page 6 of the Table of Integrals. 43 dx = x1/49- x2 14 38: Table of Integrals for basic forms and forms involving ax+b (page 1) Basic Forms 1. k dx = kx + C (any number k) 2. X' dx = 1t n + 1 + c ( n * - 1) 3. = In|x| + C 4. edx=e + c 5. adr = a Ina + C (a > 0,a # 1) 6. sinxax = -cosx + C cos x dx = sin x + C 8. sec' x dx = tan x + C 9. csc? x dx = -cotx + C 10. sec x tan xax = secx + C 11. csc x cot x dx = -csex + C 12. tan x dx = In |sec x| + C 13. cot xax = In |sin x | + C 14. sinh x dx = cosh x + C 15 . cosh x dx = sinhx + C Va _ = sina + c 17. 2 4= 18. - asecti # + c 19. di = sinh + C (a > 0) 20. dx = cosh ' + C (x> a> 0) Forms Involving ax + b 21. (ar + by" dx = (ax + byatl a(n + 1) + C, n * - 1 22. max + by dx = (ax + b) + far + b. [n+2 n+ 1 + c, n#-1, -2 23 . (ax + by'dx = hinlax + bl + c 24. max + bylax = = - 4 Inlar + b| + c 25. max + by ? dx = & Inlar + bl + ar+ 6 + C 26. nad*= 27. ( Vax + b)" dx = 2 \\Var + b) "+2 n+ 2 - + c, n#-2 28. Var + bax = 2Vax + 6+box 29. (a ) dx = 1 In Var + 6 - Vb + c (b)d vb Ivar + 6+ vol *Var 6 = tan- ax - b + c 30. ( Var + bax = - Var+ + ; a +C 31. Vax +6=- Var+ b _ bx dx + C - 25 xVar + 6 39: Table of Integrals for forms involving a^2+x^2, a^2-x^2, and x^2-a^2 (page 2) https://xlitemprod.pearsoncmg.com/api/v1/print/highered 27/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen Forms Involving a2 + x2 32. - dx - 4tan- # + C 33 . Ta + x ) = = 20' ( @2 + 1 3 ) + 20 tan 'd + c A= sinh ( + c = In(x + Va +x)+ c 35. ( Vatxax = Vato + gin(x + Vatx )+ c 36. ( 8 V a + xax = # (a + 20 ) Va +x - 4 In(x + Va+ x)+ c 37 . ( Va tedx = Vatx - al at Vato+ c [ Na tedx = In(x + Vatx) - Vat&+ c 39.dx = -9- In -2In ( x + Vatx ) + ma+ + + c -d = - Lina + votr + c RVatr a-x Forms Involving a2 - x2 12. de = Lin + +c 43. / To de 4. / Vad = sing + c 45. ( Va - x d = =Va - x+ sin-' + C 46. X Va - Fax = $sing - Java - 8(0 - 212) + C 47. / Vapidx = Va - F - alno + Vo -+ + C 48. / Va- Idx =- sing- Va -&+ C 49. vadx = 2sin'd - kra - F+ c 2Va - x Forms Involving x2 - a2 $2 . ] VT _ = In|x + Vx - al + c 53. / VP - dax = =Vr - a - $1/x + Vx - al+c AT ( V x - a )" -2 dx, n #- 1 x ( Vx - 97 ) 2-n 55. ] (VP -Q) (n - 2)a-) ( V/x2 - 2 )1-2 1 # 2 dx ( 2 - n) n+ 2 - + c, n # - 2 57. ( RV P - dax = = (27 - a ) VP - a - -In/x + Vx-al+c so. ( V& - dx = In |x + VP - al - VR - &+ c MR-d'+C 40: Table of Integrals for trigonometric forms (page 3) https://xlitemprod. pearsoncmg.com/api/v1/print/highered 28/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen Trigonometric Forms 63. sin ax dx = - 4 cos ax + C 64. / cos ax ax = a sin ar + C 65. sin' ardx = _ sin 2ax + C 4a 66. cos' ardx = + sin zar + c 67. sin" ardx = _ Sin"- arcos ax + " - 1 /sin"-2 ax dx 68. / cos" ax ax = cos"-' ar sin ax + 1 - 1 /cos"-2 ax dx 69. (a) sin ax cos bx dx = - cos(a + bjx cos(a - b) . a # 12 2 (a + b) 2(a - b) sin ax sin bx dr = sin(a - b)x sin(a + b)x DM + C, d b 2(a - b) 2(a + b) (c) cos ax cos bx dx = sin(a - b)x sin(a + b)x 2(a - b) 2(a + b) 70. sin ax cos ax dr = _ cos 20 + 4a . sin" ax cos ax ax = sin" at + C, n # -1 (n + 1 )a 72. cos ax ax = a In | sin ax| + c sin ax 73. cos" ax sin ax dx = - costtax + C. n # -1 (n + 1)a+ 74. sin ax ax = - = In | cos ax| + c a(m + n) 75. sin" arcos" ardx = _ sin' ar cost ax , " - 1 sin"-2 ax cos" ardx, n #-m (reduces sin" ax) a(m + n) 76. sin" arcos" ar dx = Sin"tax cos" " ax + m - 1 sin" ax cos"-2 ax dx, m # -n (reduces cos" ax) 77. d.x b + csinar " avplan ! Vo Etan (# - ax) + c. >2 78. dx b + c sinax ave In C + bsin ax + Ve- - bcosax + C, be 12 82. b + ccos ax dx ave In C + bcos ar + Va - b sinax + c. 12< < b + c cos ax Trigonometric Forms 41: Table of Integrals for trigonometric forms (page 4) https://xlitemprod.pearsoncmg.com/api/v1/print/highered 29/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen Trigonometric Forms 83. / 1 + cos at = atan " + c 84. 1 - cos ar = - acot " + c 85. ( xsin ax ax = sinax - acosar + C 86. / xcos ax ax = cosax + a sin ax + C 87. x* sin ax ax = - = cos ax + / x-1 cos ax dx 88. " cos ax ax = a sinax - 1 1-I sin ax ax 89. / tan ardx = = In |sec ax) + C 90. / cot ardx = =in |sin axl + C 91. tan' ax ax = =tan ax - x + C 92. cof ardx = - -cotax - x + C 93. / tan" ax ax - tan' ' ox - tank ? ardx, n # 1 94. cor" ax ax = - a(n - 1) - com -2 ardx, n # 1 95. sec ax ax = = In |sec ar + tan ax| + C 96. csc ar dx = - 4 In |csc ax + cot axl + C 97. sect ax dx = atan ax + C 98. csco ax dx = - acotar + C a(n - 1) 99. sec" ax ax = sec"-? ax tan ax + 1 - 2 / sec"-2 ax dx, n # 1 100. csch ax ax = - csc*-2 ar cotax + n - 2 / esca-2 ax dx, n # 1 a(n - 1) 101. sec" ax tan ardx = Sec at + C, n # 0 102. csc" ax cot ax dx = _ cc" at + C, n # 0 42: Table of Integrals for inverse trig forms, exponential/logarithmic forms, and forms involving square root of 2ax-x^2, a>0 (page https://xlitemprod.pearsoncmg.com/api/v1/print/highered 30/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen Inverse Trigonometric Forms 103. sin-' axax = xsin lax + 4VI - ax + c 104. cos ' ardx = xcos' ax - 2VI - der + C 105. tan ' ardx = xtan ' ax - 2 In (1 + d'x ) + C 106. X" sin ' ardx = 1" n + [ sin lax - 7 4 1 / V1 -ax n * -1 107. ( x" cos ' ardx = anti = n + [ cos ' ax + 7 4 1 / x x , n # - 1 VI - or 108. " tan ' ardx = path got l dx 2, n # -1 Exponential and Logarithmic Forms 109. edx = be + c 110. ba dx = 1 bex alnb + C, b > 0,b = 1 In1. fxedx = = (ar - 1) + C 112. red = are - reax 113. xbox ax = Amber alnb - ainb * 1 dx. b > 0.b = 1 114. e sin by dx = 2 (a sin bx - bcos bx) + C 115. e cos bx dix =- 2 + be (@ cos bx + bsin bx) + C 116. Inaxax = xInax - x + C 117. x" (In ax)" dx = Int (In ary n + 1 7 + 1 / x"(In ary -lax , n # - 1 118. '(In ax)" dx = In aryatl m+ 1 + c , m # - 1 119. - dy = In [In ax) + C Forms Involving V2ax - xz, a > 0 -dx = sin-1 : -") +c 121. / Vzax - x3 dx = 1 - Vzax - 8 + $ sin1 (1 - ") + c 122. (Vzar - 3)" dx - (x - a)(V2ax - 13)" n + 1 n+ 1 (V 2ax - x2) "-2 dx (x - a)(V2ax - )2-" 123. (zar - 12)" (n - 2) + n - 3 (n - 2x7) ( Vzar - 12 )"- dx 124. XVzax - x' dx - ( + @)(2x - 3a)V2ax - x + 9 sin-1 ( 1 - a ) + c 125. V2ax - & dx = Vzax - x3 + asin-1 (1 - ") + c 126. / V2ax - dx = -2 /20 - sin-1 (1 2 ") + c 127. Van = asin- ("- ") - V2ax - x + C 128./ 43: Table of Integrals for hyperbolic forms (page 6) https://xlitemprod.pearsoncmg.com/api/v1/print/highered 31/61\f7/4/23, 11:16 PM Homework 8.7-Anh Nguyen Use a table of integrals to determine x ex dx. Click here to view page 1 of the integral table. " Click here to view page 2 of the integral table.45 Click here to view page 3 of the integral table. 46 Click here to view page 4 of the integral table.47 Click here to view page 5 of the integral table." Click here to view page 6 of the integral table.49 x3 8X dx = 4096 (512e8Xx3 - 3(648Xx2 -2(8e8Xx-8x) ) ) + c 44: Table of Integrals (page 1) Substitution Rule g (b ) J Ha(x)'(x) ax= Ji(u) du (u = g(x) Ha(x)g' (x ) ax = ] fu ) du g(a) Integration by Parts Judv=ur- frau fur ax=uvla - Svu ox Basic Integrals | x" dx = x n+ 7 X " + 1 + C;n# - 1 J = In / x/ + c cosax dx = - sin ax + C sin ax dx = - - cosax + C tanx dx = In |sec x| + c cotx dx = In | sin x | + C secx dx = In /secx+ tan x| + c csc x dx = - In /cscx + cotx| + C Jeax ax = =eax + c bax dx = = 1 alnb - bax + C; b > 0, b#1 In x dx = x In x - x + C log , X dx = In ( X Inx - x ) + C dx = sin - 1_ + c dx = 1 tan - 12 + c dx 12 +2 2 - sec -1 ~ +c a a a x1x2 - a2 a sin - 'x dx = xsin -'x + $1-x2 + c cos - 'xdx=xcos -1x- 1-x + c tan 'x dx =xtan -1x- In ( sec 1x dx = xsec- 1x - In (x + Vx2 - 1 ) + c sinh x dx = cosh x + C cosh x dx = sinh x + C sech 2 x dx = tanhx + C csch 2x dx = - coth x + C sech xtanh x dx = - sechx + ( csch x coth x dx = - cschx + C tanh x dx = In coshx + C coth x dx = In |sinh x| + C sech x dx = tan " sinh x + C= sin -1 tanh x + C cschx dx = In tanh + c 45: Table of Integrals (page 2) Trigonometric Integrals https://xlitemprod.pearsoncmg.com/api/v1/print/highered 33/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen cos 2 x dx = 2* sin 2x +C sin 2x dx = sin 2x 4 + C 4 sec 2 ax dx = ~ tan ax + C csc 2 ax dx = - - cotax + C tan 2 * dx = tan x - x + C cot 2 x dx = - cotx - x + C |cos 3 x dx = - = sin 3x + sinx + C sin 3 x dx = = cos 3x - cos x + C sec 3 x dx = = sec x tanx - csc 3 x dx= - > csexcotx - > In /cscx+ cotx| + C tan 3 x dx = - tan 2x - In /secx| + c cot 3 x dx = - cot 2x - 11 sec " ax tan ax dx = = sec "ax + C; n#0 csc " ax cotax dx = - macscax + C; n#0 dx 1 + sin ax - tan dx dx dx 1 - sin ax a 1tan II , ax + c - tan ax 1 + cos ax a ax + C 1 - cos ax a 2 sin mx cos nx dx = - cos (m + n)x cos (m - n)x - + C; m2 #n2 2(m + n) 2(m - n) sin mx sin nx dx = sin (m - n)x sin (m + n)x 2(m - n) 2(m + n) + C; m2 # n2 cos mx cos nx dx = sin (m - n)x sin (m + n)x 2(m - n) 2(m + n) + C; m2 # n2 46: Table of Integrals (page 3) Reduction Formulas for Trigonometric Functions cos "x dx = - cos "- 1xsinx+ "-1 / cos -2x dx sin "x dx= - _ sin "- 1xcosx+ 2-1 / sinn-: tan "'x dx = tan " -1x 1 - 1 J tan "-2 x dx; n # 1 | cot "x dx = _ cot " - 1x n - 1 cot " -2 x dx; n # 1 sec " x dx = S sec "- 2xtan x _n-2 [ s n - 1 n - sec " - 2 x dx; n # 1 csc"xdx= _ CSC"-2xcotx n-2 0 n - 1 csch - 2 sin " xcos "x dx = _ s sin m - 1 x xcos "+ 1x m-1 m +n m + nJ sin " - 2 xcos " x dx ; m # -n sin xcos "x dy = sin m+ 'xcos - 1x - n-1 m + n m + nJ sin " xcos " - 2 x dx; m# -n x" sin ax dx = - x" cosax n n-1c a cos ax dx; a # 0 |x " cos ax dx = - x" sin ax a - x - 1 s sin ax dx; a Integrals Involving a2 - x2; a > o [ Vaz -x2 dx = $ Vaz - x2 + asin-12 + c dx a + Va2 -x2 2 - 2 In xva2 - x 2 + C a dx Va2 - x2 x2 Na2 - x2 + C a x [x2 Na? - x2 dx = (2x2 -a? ) Val - x2 + sin-12 + c [Vaz -x2 x2 * 2 dx = - - Na2 -x2 + C- sin-1-+ c 7 dx = - -Va2 -x2 + sin -1- + c a2 a Va 2 - x 2 2 12_ x2 = =_ In x-a 2a 47: Table of Integrals (page 4) https://xlitemprod.pearsoncmg.com/api/v1/print/highered 34/617/4/23, 11:16 PM Integrals Involving x2 - a2; a > o Homework 8.7-Anh Nguyen ( x2 -22 ax = * Vx - a2 - , In x + 1x2 -a? |+ c dx = In / x + 1x2 - 22 | + c dx Vx2 -a2 x 2 N x 2 - a2 + C ax [ x x - 27 Ox = * (2x2 - 2 2 ) 1x - 2" - 2 in/ x + 1x-a?| +c (vx2 - a2 - dx = In / x + x2 - a21- Vx2 - a2 - +C x2 X VX 2 - 2 2 5 dx = = In/ x + Vx2 -a? | + x x2 -a2 + c dx 2 - 2 2 - 2a dx x ( x 2 - 2 2 ) = In x2 - 2 + c Integrals Involving a2 + x2; a > o [ Vaz + x2 ax = > a2 +x2 + 3-In(x+Val +x? ) +c dx = = In (x + Va2 +x2 ) + c dx - In a - Val +x2 a + C dx Va2 + x2 xvaz+ X *2 Na2 + x 2 + C a x [ x2 Vaz +x2 dx= x (a2 + 2x2) Va2 +x2- 2 in (x+ a2+x2 ) + c [ Va +x Va+ x 2 dx = In x + Va2 + x21. + C x 2 dx = - - In (x+ Vaz +x2)+ xvaz+x2 2 . + C ( Vaz + x 2 - ax = 12 2 + x 2 - aln a+ Va2 +x2 X + C dx (a2 +x2) 3/2 = a? Va2 +x2 + C dx * (a2 +x2 ) In C 2a 2 48: Table of Integrals (page 5) Integrals Involving axb; a #0, b > 0 ( ax + b ) " dx = - ( ax + b) n + 1 a (n + 1) - + C; n# - 1 ( Vax + b ) " dx = 2 ( vax + b ) n + 2 n+2 - + C; n# -2 dx 2 - tan - ax - b + C dx xvax - b Vax + b - vb Vb b x vax + b = In Vb I Vax + b + Vb + C X dx = - b In lax + b| + C ax + b a dx = - 1 ax + b 273 ((ax + b)< - 4b(ax + b) + 2b2 in lax + b/) + c dx a In |ax + b x (ax + b) bx 6 2 X + c x Vax + b dx = 2 , 15a 2 (3ax - 2b)(ax + b)3/2 + c X dx = - 2 ax + b 2 (ax - 2b ) vax + b + C x(ax + b) " dx = (ax + bin + 1 ( ax + b b 2 2 n +2 n + 1 + C; n# -1, -2 dx x (ax + b ) In X ax + b + C Integrals with Exponential and Trigonometric Functions https://xlitemprod. pearsoncmg.com/api/v1/print/highered 35/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen eax sin bx dx = ex(a sin bx - b cos bx) ed(a cos bx + b sin bx) + C eax cos by dx = + C a- + b a-+ 62 49: Table of Integrals (page 6) Integrals with Exponential and Logarithmic Functions dx y = In |Inx| + c *n+ 1 xinx 1 + 1 1 - 1 + 1 ) + Cin# - 1 xex ax = * * - * + c J xmeax ax = _xreax _ " fxn-1eax dx; a*0 in "x dx =xIn "x-n/ In"-1xox Miscellaneous Integrals [x" cos -'xax= 4 x+1 cos-1x+ [ x" dx 1in* - 1 [x" sin - "xox = net x't 'sin -1x-[X 1+1 dx in* - 1 [x" tan - 'x dx=1 xtitan -1 x- J-2 a in* - 1 Vzax - x2 dx= X-a J = 2 V2ax - x2 + 2 si + 82 sin -1 (x2 8 + C;a>0 dx V 2ax - x2 2+ C; a > 0 https://xlitemprod.pearsoncmg.com/api/v1/print/highered 36/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen 10. Use a table of integrals to evaluate the following indefinite integral. The integrals may require preliminary work, such as completing the square or changing variables, before it can be found in a table. x2 + 10x+41 -dx Click here to view page 1 of the Table of Integrals. 50 Click here to view page 2 of the Table of Integrals. 51 Click here to view page 3 of the Table of Integrals.32 Click here to view page 4 of the Table of Integrals.5 Click here to view page 5 of the Table of Integrals. Click here to view page 6 of the Table of Integrals.5 x2 + 10x +41 dx = - tan -1 (* + 5 ] + c 50: Table of Integrals for basic forms and forms involving ax+b (page 1) Basic Forms 1. k dx = kr + C (any number k) 2. X" dx = 1+1 nti + c (n * - 1 ) 3. = In|x| + C 4 ed=etc 5. adx = + C (a > 0,a# 1) 6. sin xax = -cosx + C 7. cos x dx = sin x + C 8. sec' x dx = tan x + C 9. csco x dx = -cotx + C 10. sec x tan x dx = sec.x + C csc x cot x dx = -csc x + C 12. tan x dx = In |sec x| + C cot x dx = In |sin x| + c 14 . sinh x dx = coshx + C 15. cosh x dx = sinh x + C 16. / Vaz = Sin 'd + c 18 . - dsee- # + c Va ty = sinh' a + c (a> 0) 20 . V = cosh -1 4 + C (x>a > 0) Forms Involving ax + b 21. (ar + by" dx = (at + byatl a(n + 1 ) + C, n * - 1 22. max + by dx = (ar + 5jut [ ar + b _ 6 + c. n # - 1, - 2 [n+2 - 23. (ar + by ' dx = =Inlax + b / + c 24. ( max + by ' dx = = - 4 Inlar + b| + c 25. max + by ? dx = + Inlax + bl + as " 6+ ax + 6 + c 26 . / Hat + 6 )= 27. ( Var + b)" dx = 2 (Vax + 6) +2 n+ 2 + C, n#-2 28. Var + bax = 2Vax + 6+box 29. (a) dix = 1 In Var + b - Vb + C (b) -dx = 2- tan-1 ax - b + c "Ivax + 6 + vol Var+bax = _ Var+ + ad + c 31. / x Var+6 Var+b _ bx =+c 26 / WVax + 6 51: Table of Integrals for forms involving a^2+x^2, a^2-x^2, and x^2-a^2 (page 2) https://xlitemprod.pearsoncmg.com/api/v1/print/highered 37/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen Forms Involving a2 + x2 32. - dx - 4tan- # + C 33 . Ta + x ) = = 20' ( @2 + 1 3 ) + 20 tan 'd + c A= sinh ( + c = In(x + Va +x)+ c 35. ( Vatxax = Vato + gin(x + Vatx )+ c 36. ( 8 V a + xax = # (a + 20 ) Va +x - 4 In(x + Va+ x)+ c 37 . ( Va tedx = Vatx - al at Vato+ c [ Na tedx = In(x + Vatx) - Vat&+ c 39.dx = -9- In -2In ( x + Vatx ) + ma+ + + c -d = - Lina + votr + c RVatr a-x Forms Involving a2 - x2 12. de = Lin + +c 43. / To de 4. / Vad = sing + c 45. ( Va - x d = =Va - x+ sin-' + C 46. X Va - Fax = $sing - Java - 8(0 - 212) + C 47. / Vapidx = Va - F - alno + Vo -+ + C 48. / Va- Idx =- sing- Va -&+ C 49. vadx = 2sin'd - kra - F+ c 2Va - x Forms Involving x2 - a2 $2 . ] VT _ = In|x + Vx - al + c 53. / VP - dax = =Vr - a - $1/x + Vx - al+c AT ( V x - a )" -2 dx, n #- 1 x ( Vx - 97 ) 2-n 55. ] (VP -Q) (n - 2)a-) ( V/x2 - 2 )1-2 1 # 2 dx ( 2 - n) n+ 2 - + c, n # - 2 57. ( RV P - dax = = (27 - a ) VP - a - Cin/x + Vx-al+c so. ( V& - dx = In /x + VP - al - VR - &+ c MR-d'+C 52: Table of Integrals for trigonometric forms (page 3) https://xlitemprod.pearsoncmg.com/api/v1/print/highered 38/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen Trigonometric Forms 63. sin ax dx = - 4 cos ax + C 64. / cos ax ax = a sin ar + C 65. sin' ardx = _ sin 2ax + C 4a 66. cos' ardx = + sin zar + c 67. sin" ardx = _ Sin"- arcos ax + " - 1 /sin"-2 ax dx 68. / cos" ax ax = cos"-' ar sin ax + 1 - 1 /cos"-2 ax dx 69. (a) sin ax cos bx dx = - cos(a + bjx cos(a - b) . a # 12 2 (a + b) 2(a - b) sin ax sin bx dr = sin(a - b)x sin(a + b)x DM + C, d b 2(a - b) 2(a + b) (c) cos ax cos bx dx = sin(a - b)x sin(a + b)x 2(a - b) 2(a + b) 70. sin ax cos ax dr = _ cos 20 + 4a . sin" ax cos ax ax = sin" at + C, n # -1 (n + 1 )a 72. cos ax ax = a In | sin ax| + c sin ax 73. cos" ax sin ax dx = - costtax + C. n # -1 (n + 1)a+ 74. sin ax ax = - = In | cos ax| + c a(m + n) 75. sin" arcos" ardx = _ sin' ar cost ax , " - 1 sin"-2 ax cos" ardx, n #-m (reduces sin" ax) a(m + n) 76. sin" arcos" ar dx = Sin"tax cos" " ax + m - 1 sin" ax cos"-2 ax dx, m # -n (reduces cos" ax) 77. d.x b + csinar " qv/plan ! Voctan (# - ax) + c. >2 78. dx b + c sinax ave In C + bsinax + Ve- - bcosax + C, be 82. b + ccos ax dx ave In C + bcos ar + Va - b sinax + c. 12< < b + c cos ax Trigonometric Forms 53: Table of Integrals for trigonometric forms (page 4) https://xlitemprod.pearsoncmg.com/api/v1/print/highered 39/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen Trigonometric Forms 83. / 1 + cos at = atan " + c 84. 1 - cos ar = - acot " + c 85. ( xsin ax ax = sinax - acosar + C 86. / xcos ax ax = cosax + a sin ax + C 87. x* sin ax ax = - = cos ax + / x-1 cos ax dx 88. " cos ax ax = a sinax - 1 1-I sin ax ax 89. / tan ardx = = In |sec ax) + C 90. / cot ardx = =in |sin axl + C 91. tan' ax ax = =tan ax - x + C 92. cof ardx = - -cotax - x + C 93. / tan" ax ax - tan' ' ox - tank ? ardx, n # 1 94. cor" ax ax = - a(n - 1) - com -2 ardx, n # 1 95. sec ax ax = = In |sec ar + tan ax| + C 96. csc ar dx = - 4 In |csc ax + cot axl + C 97. sect ax dx = atan ax + C 98. csco ax dx = - acotar + C a(n - 1) 99. sec" ax ax = sect-? ax tan ax + 1 - 2 /sec"-2 ardx, n # 1 100. / csch ax ax = - csc"-2 ar cotax + n - 2 / esca-2 ax dx, n # 1 a(n - 1) 101. sec" ar tan ardx = Sec at + C, n # 0 102. csc" ax cot ax dx = _ cc" at + C, n # 0 54: Table of Integrals for inverse trig forms, exponential/logarithmic forms, and forms involving square root of 2ax-x^2, a>0 (page https://xlitemprod.pearsoncmg.com/api/v1/print/highered 40/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen Inverse Trigonometric Forms 103. sin-' axax = xsin lax + 4VI - ax + c 104. cos ' ardx = xcos' ax - 2VI - der + C 105. tan ' ardx = xtan ' ax - 2 In (1 + d'x ) + C 106. X" sin ' ardx = 1" n + [ sin lax - 7 4 1 / V1 -ax n * -1 107. ( x" cos ' ardx = anti = n + [ cos ' ax + 7 4 1 / x x , n # - 1 VI - or 108. " tan ' ardx = path got l dx 2, n # -1 Exponential and Logarithmic Forms 109. edx = be + c 110. ba dx = 1 bex alnb + C, b > 0,b = 1 In1. fxedx = = (ar - 1) + C 112. red = are - reax 113. xbox ax = Amber alnb - ainb * 1 dx. b > 0.b = 1 114. e sin by dx = 2 (a sin bx - bcos bx) + C 115. e cos bx dix =- 2 + be (@ cos bx + bsin bx) + C 116. Inaxax = xInax - x + C 117. x" (In ax)" dx = Int (In ary n + 1 7 + 1 / x"(In ary -lax , n # - 1 118. '(In ax)" dx = In aryatl m+ 1 + c , m # - 1 119. - dy = In [In ax) + C Forms Involving V2ax - xz, a > 0 -dx = sin-1 : -") +c 121. / Vzax - x3 dx = 1 - Vzax - 8 + $ sin1 (1 - ") + c 122. (Vzar - 3)" dx - (x - a)(V2ax - 13)" n + 1 n+ 1 (V 2ax - x2) "-2 dx (x - a)(V2ax - )2-" 123. (zar - 12)" (n - 2) + n - 3 (n - 2x7) ( Vzar - 12 )"- dx 124. XVzax - x' dx - ( + @)(2x - 30)V2ax - x + 9 sin-1 ( 1 - a ) + c 125. V2ax - & dx = Vzax - x3 + asin-1 (1 - ") + c 126. / V2ax - dx = -2 20 - sin-1 (1 2 4) + c 127. = asin" (" - 4) - Vzax -X + C 128. /- 55: Table of Integrals for hyperbolic forms (page 6) https://xlitemprod.pearsoncmg.com/api/v1/print/highered 41/61\f7/4/23, 11:16 PM Homework 8.7-Anh Nguyen 11. Use a table of integrals to evaluate the following indefinite integral. The integrals may require preliminary work, such as completing the square or changing variables, before it can be found in a table. dx x 2 - 4X =, x > 4 Click here to view page 1 of the integral table.3 Click here to view page 2 of the integral table.5 Click here to view page 3 of the integral table.5 Click here to view page 4 of the integral table. 59 Click here to view page 5 of the integral table. Click here to view page 6 of the integral table.1 dx x2 -4X = = In |x- 2+ (x-2)2 - 4| +c 56: Table of Integrals (page 1) Substitution Rule g (b ) (1(9(x)g'(x ) ax = Jiu ) du ( u = 9( x ) [ng(x)g'(x) ax= | +(u) du a g(a ) Integration by Parts Juav=uv- Svau fur' ax = uvla- Svu'dx Basic Integrals x" dx = -xn+1 + C;n# - 1 = In / x | + c cos ax dx = - sin sin ax dx = - - cos ax + C tan x dx = In |sec x| + C cot x dx = In |sin > sec x dx = In / sec x + tan x| + C csc x dx = - In |csc x + cot x| + C eax dx = - eax + c bax dx = - 1 albb + c; b > 0, b # 1 Inx dx = x Inx - x + C log b X dx = = Inb ( X dx 5 = sin - 12 + c dx Va2 - x 24 23 - tan -14 + c se xVx - sin -'xox =xsin 'x +1 1-X" +c cos -'x dx =xcos -1x- 1-x2 +c tan - 1x dx = x tan sec -1x dx = xsec-1x - In (x + x2 - 1 ) +c sinh x dx = coshx + C cosh x dx = sinh > sech 2 x dx = tanh x + C csch 2x dx = - coth x + C sech x tanh x dx = csch x coth x dx = - cschx + C tanh x dx = In cosh x + C coth x dx = In sin sechx dx = tan sinhx + C= sin ' tanh x + C |csch x dx = In tann 2 + c 57: Table of Integrals (page 2) Trigonometric Integrals https://xlitemprod.pearsoncmg.com/api/v1/print/highered 43/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen |cos 2 x dx = 2 + - sin 2x 4 + C sin 2x dx = 2 sin 2x 4 + C sec 2 ax dx = - csc 2ax dx = - ~ cotax + C tan 2 x dx = tan x - x + C [ cot 2 x dx = - cos ' x dx = - = sin 'x + sin x + C sin 'x dx = = cos x- cos x + C sec 3 x ax = } csc 3x dx = - - csexcotx- - In csex+ cotx/ +C tan 3 x dx = - tan 2x - In /secx| + c cot 3 x dx = - sec " ax tan ax dx = = - sec "ax + C ;n # 0 csc " ax cot ax dx = - Csc "ax + C ;n #0 1 + sin ax dx - tan - ax + c dx ax 1 - sin ax 1 + cos ax - tan ? + C 1 - cos ax cos (m + n)x cos (m - n)x sin mx cos nx dx = 2(m + n) - + C;m # n2 2(m - n) sin (m - n)x sin (m + n)x sin mx sinnx dx = 2(m - n) + C; m2 # n2 2(m + n) cosmx cos nx dx = sin (m - n)x sin (m + n)x 2(m - n) 2(m + n) -+ C ;m # n2 58: Table of Integrals (page 3) Reduction Formulas for Trigonometric Functions |cos "x dx = - cos "-'xsinx+ "- cos "-2x dx sin "'x dx = - 1 sin "- 1xcosx+ ". tan' tan " x dx = - nn -1, n - 1 tan " - 2 x dx ; n # 1 cot " x dx = _ cot " - 1x 1 - 1 sec "x dy = sec "-2x tan x n - 2 n- 1 n - 1 .) sec " -2x dx; n #1 csc "x dx = _ csc"-2x cotx n- 1 sin " x cos "x dx = - sin -1x cos +1x m - 1 m + n m +n.J sin m- 2 xcos " x dx ; m # - n sin "x cos "x dx = sin m+ 1x xcos" - 'x n -1 m + n m + n J sin "xcos "-2 x dx; m# -n x" sin ax ax = _ x" cos ax x" sin ax a x " - cos ax dx ; a # 0 [ x" cos ax dx = - a Integrals Involving a - x2; a > 0 https://xlitemprod.pearsoncmg.com/api/v1/print/highered 44/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen [ Vaz - x2 ax=>Va2-x2 + sin-12+ dx a + Va2 -x2 a + C - In xval -> X + C dx Va2 - x2 x2 Na2 -x2 + C [x2 Ja2 - x2 dx = = (2x2 -a?) a2 -x2 + 3 sin 12 +c Va2 -x2 dx = - - va2 - x2 + C- sin-1_ + c x2 X a 2 dx= - 2 Va2 - x2 + 3 sin-1X + dx - In 2a 59: Table of Integrals (page 4) Integrals Involving x2 - a2; a> 0 [ Vx2 - 2 2 Ox = * Vx2- 22 - 3- In / x + 1x2 - a? | + c JJ2 2 = In x +1x - 2 2 | + c dx Vx 2 - 22 + C x vx -a a x [ x2 1x2 - 27 Ox= (2x2-2 2) 1x2- 27 - in/x+ 1x2- 27/+1 Vx2 - 2 2 dx = In x + 1x2 - a21. Vx2 - 22 + C X Vx 2 - a2 2 dx = 2 In x + Vx2 -a2 | + x x2 -a2 + c dx 2 - 2 2 = - In X - a + c dx x ( x 2 - a2 ) 1 In x2 -a2 2a * 2 + C Integrals Involving a2 + x2; a> 0 ( vaz + x 2 Ox = = Vaz + x7 + 3- In(x+122 +x] ) +c dx Va 2 + x = = In (x + Va2 +x2 ) +c dx - In a - Val +x2 + C dx xva2 + , x Na 2 + x 2 1 2 2 2 +x 2 + C a x [ x2 a2 +x2 ax= * (22+ 2x2 ) a2 +x2- 2- in (x+ Va2 +x? ) + c ( Na2 +x2 ax = In /x + a2 +x2| va2 +x2 + C X x2 dx = - 2 In (x + Va2+x2) + xva2 +x2 a2 + x2 2 .+ C * *- dx = val + x 2 - aln a + Vaz +x2 + C dx X (a2 + x2) 3/2 2 2 2 2 + x 2 + C dx x ( 2 2 + x 2) In 2a2 + C 60: Table of Integrals (page 5) Integrals Involving ax = b; a#0, b > 0 https://xlitemprod.pearsoncmg.com/api/v1/print/highered 45/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen (ax + b)" dx = (ax + b) n + 1 a(n+ 1) - + C; n# - 1 ( Vax + b ) " dx = 2 ( vax + b ) n+ 2 n +2 - + C; n# -2 dx 2 -1 ax - b dx Vax + b - vb xvax - b tan Vb - + C x vax + b = In Vb Vax + b + vb + C X - dx = - - In lax + b| + c ax + b a dx = - ax + b 273 ( (ax +b)< - 4b(ax + b) +2b2 in lax + bl) +c dx a -+ - ax + b In x (ax + b) bx x + C xVax + b dx = 2 (3ax - 2b)(ax + b)3/2 + c 15a X 2 dx = - ax + b 32 2 (ax - 2b) vax + b + C | x ( ax + b ) " dx = ( ax + b ) " + 1 [ ax + b b 2 n + 2 n + 1 + C; n# -1, -2 dx x (ax + b ) - In X ax + b +C Integrals with Exponential and Trigonometric Functions eax sin bx dx = e (a sin bx - bcos bx) + C a + 62 eax cos bx dx =- e(a cos bx + b sin bx) + C a + 62 61: Table of Integrals (page 6) Integrals with Exponential and Logarithmic Functions dx xin x = In |In x| + c [x" Inxox= n+1 Inx - n+1 + C;n# - 1 [xe* dx= xex - ex + c Ix" ax ax = 1x"eax _ " fx - lax dx; a #0 In "x dx =xin "x-n In"-1x dx Miscellaneous Integrals x" cos 'xax= 1 x+ 1 cos-1x+ [x *n+ 1 dx Fin# -1 [ x" sin - 'x ax = n+7 xnt 'sin -1x - JX ; n* - 1 x " tan - 1x x = -xn+ 1 dx 1+ 7 *0+ 1 tan - 1 x - J -2 4 1 in * - 1 2ax - x2 dx = X-a zax-x2 + 3- sin-1 X-2 +c;a>o dx = = sin " -1 (X- 2) (2ax - x2 YOU ANSWERED: (:1x2 - AX + X - 21 + 0 https://xlitemprod.pearsoncmg.com/api/v1/print/highered 46/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen 12 3 Use a table of integrals to determine dt. This integral requires preliminary work before it can be found in a table. 136 + 16e2t Click here to view page 1 of the integral table. Click here to view page 2 of the integral table. 63 Click here to view page 3 of the integral table. Click here to view page 4 of the integral table. 65 Click here to view page 5 of the integral table. Click here to view page 6 of the integral table. 67 3t 2e' + 142 2+ 9 36+ 162t dt = 32 2e' V4e2 +9 -9 In 3 + c 62: Table of Integrals (page 1) Substitution Rule g (b ) | +(9(x)g' (x) ax = [i(u) du (u = g( x ) Ha(x)9' (x ) ax = =(u ) du g(a) Integration by Parts Judv=uv-Svau fuvax= uvla- Svu'dx Basic Integrals [x" dx = =1 x1 + 1 + c;n# - 1 fox = In/ x/ + c cosax dx = - sin ax + C sin ax dx = - - cos ax + C tanx dx = In /sec x | + C cotx dx = In |sin x| + C sec x dx = In / sec x + tan x/ + C csex dx = - In /cscx + cotx/ + c Jeax ax = Leax + c bax dx = 1 - alnbbax + c; b > o, b # 1 In x dx = xInx - x + C 109 5X x = In b ( x Inx - x ) + C dx dx Va 2 - x 2 = = sin - 1_ + c = - tan - 12 + c a a a x 1x2 - a2 - a = - sec -1 + |sin - 'x dx =xsin -1x+ $1-x2 +c cos - 1x dx = xcos -1 x - V1-x2 + c tan - 1x dx =xtan -1x - In sec - 1x dx =xsec-1x- In (x + 1x2 - 1) +c sinh x dx = cosh x + C cosh x dx = sinhx + C sech 2 x dx = tanh x + C csch 2 x dx = - coth x + c sech x tanh x dx= - sechx + csch x coth x dx = - csch x + C tanh x dx = In cosh x + C coth x dx = In | sinh x| + c sech x dx = tan - 1 sinh x + C = sin -1 tanh x + C | csch xdx = In tank 2 + c 63: Table of Integrals (page 2) Trigonometric Integrals https://xlitemprod.pearsoncmg.com/api/v1/print/highered 47/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen cos 2 x dx = 2* sin 2x sin 2x 4 +C sin 2x dx = + C 4 sec 2 ax dx = = tan ax + ( csc 2 ax dx = - - cotax + C tan 2 * dx = tan x - x + C cot 2 x dx = - cotx - x+ |cos 3 x dx = - 7 sin 3x + sinx + C sin 3 x dx = = cos 3x - cos x + C sec 3 x dx = = sec xtan> csc 3 x dx= - > csexcotx - > In /cscx+ cotx| + C tan 3 x dx = - tan 2x - In /secx| + c cot 3 x dx = - 7 cot 2x - sec " ax tan ax dx = = sec "ax + C; n#0 csc " ax cotax dx = - macscax + C; n#0 dx 1 + sin ax - tan dx dx ax + C ax 1 - sin ax a 1tan II , ax + c dx 1 + cos ax - tan a -c 1 - cos ax a 2 sin mx cos nx dx = - cos (m + n)x cos (m - n)x - + C; m2 #n2 2(m + n) 2(m - n) sin mx sin nx dx = sin (m - n)x sin (m + n)x 2(m - n) 2(m + n) + C; m2 # n2 cos mx cos nx dx = sin (m - n)x sin (m + n)x 2(m - n) 2(m + n) - + C; m2 # 12 64: Table of Integrals (page 3) Reduction Formulas for Trigonometric Functions cos "x dx = _ cos"- 1xsinx+ "-1 / cos -2x dx sin "x dx = - _ sin "-1xcosx+ 2-1 / sinn- tan "'x dx = tan " -1x n - 1 J tan "-2 x dx; n # 1 cot "'x dx = _ cot " - 1x n - 1 cot " - 2 x dx; n # 1 sec " xdx = s sec"- 2xtan x n-2 [ n- sec " - 2 x dx ; n # 1 csc " x dx = _ csc " -2x cotx n -2 1 - 1 n -1 1 - 1 ) csc - sin "xcos "x dx = _ sinm-1x 'xcos "* 1x m -1 m +n m +nJ sin m-2 xcos "x dx; m# -n sin "xcos "x dy sin "+ 1xcos - 1x n - 1 m + n m + nJ sin " xcos " - 2 x dx; m# -n x" sin ax dx = - x" cosax n n-1 x " cos ax dx = - x" sin ax a cos ax dx; a # 0 a - x - 1 s sin ax dx; Integrals Involving a2 - x2; a > o [ Vaz -x2 dx = $ Vaz - x2 + asin-12 +c dx a + Va2 -x2 2 - - In + C xva2 - x 2 a dx Va2 - x2 + C x2 Na2 - x2 [x2 Na? - x2 dx = (2x2 -a? ) Val - x2 + sin-12 + c a x [Vaz -x2 * 2 dx = - - a2 -x2 + C- sin-1- + c x2 Va 2 - x 2 7 dx = - -Va2 -x2 + sin -1- + c a2 a 2 12_ x2 = =~ In x-a 2a 65: Table of Integrals (page 4) https://xlitemprod. pearsoncmg.com/api/v1/print/highered 48/617/4/23, 11:16 PM Integrals Involving x - a ; a > o Homework 8.7-Anh Nguyen Vx2 - 22 dx = $ 1x2 - 22 - 2- In/ x + 1x2 -a2| +c dx 1x 2 -2 = In x + 1x 2 - 2? | + c dx x2 - 22 x 2 NX 2 - + C ax [ x2 1x2 - a2 dx= (2x2 - 2? ) 1x2 - 27 - in/ x + 1x- 2 2| +c Wx2 -a2 - dx = In /x + 1x2 -a21 Vx2 - a2 + C 2 2 dx = = in/ x + 1x - 2?| +2 1x - 2 2 + c J2-22 = 22 10 xtal dx (2 - 2 2 ) 22? |x2 - 2 + c Integrals Involving a2 + x2; a > o [ Vaz + x 2 Ox = $ Vaz + x 2 + 3- In (x+ Na2 +x2 ) + c dx Vaz + x2 = In (x + Vaz +x2 ) + c dx -=1 a- Va2 +x2 Va2 + x 2 *va2 + x2 a + c dx + C x2 Na2 +x2 [ x2 Vaz + x2 dx = * (a2+2x2 ) Va2 + x2 - 3- In (x + 1a2 +x2 ) + c [ Vaz + x2 dx = In x + Va2 +x21. Vaz+ x2 2 C x2 2 2 + x 2 dx = - - In (x + Vaz + x2 ) , xVa2 +x2 2 + C [Na2 + x2 2 x dx = Va2 +x2 -aln a + Va2 +x2 X + C dx (a2 +x2) 3/2 a2 Naz + x2 * C dx -In x2 * ( 2 2 + * 2 ) 2a2 2 2 + x 2 + C 66: Table of Integrals (page 5) Integrals Involving ax + b; a # 0, b > 0 (ax + b ) " dx =- (ax + b)n + - + C; n# -1 J ( Vax + 5 ) " dx = = *= 2 ( Vax + b ) +2 a (n + 1) n + 2 - + C; n# -2 dx 2 tan - 1 ax - b - + C dx b = In Vax + b - Vb xvax- b Vb x vax + b Vb Vax +6 + vb + C dx = _ _ ax + b a 7 In lax + b| + c (x2 ax + b 2a 3 - dx = - (lax + b) 2 - 4b (ax + b ) + 262 in lax + b / ) + c dx a x2 (ax + b) In ax + b | X + C bx |xax + b dx= 2 1522 7 (3ax - 2b)(ax +b)3/2 + c X 2 ~ dx = - Vax + b -(ax - 2b) vax + b + C x ( ax + b ) " dx = ( ax + b ) " + 1 ax + b a 2 n + 2 n + 1 + C ; n # - 1 , - 2 dx In X x(ax + b) ax + b + C Integrals with Exponential and Trigonometric Functions eax sin bx dx= eax (a sin bx - bcos bx) a2 + 62 - +C eax cos by dx = eax (a cos bx + b sin bx) - +c a + 62 67: Table of Integrals (page 6) Integrals with Exponential and Logarithmic Functions https://xlitemprod.pearsoncmg.com/api/v1/print/highered 49/61\f7/4/23, 11:16 PM Homework 8.7-Anh Nguyen 13. Use a table of integrals to find the area of the region bounded by the graph of y = and the x-axis between x = 0 and x = 2. x2 - 10x+26 Click here to view page 1 of the integral table. Click here to view page 2 of the integral table.69 Click here to view page 3 of the integral table. " Click here to view page 4 of the integral table. Click here to view page 5 of the integral table. " Click here to view page 6 of the integral table. Set up the integral that gives the area of the region. Use increasing limits of integration. Select the correct choice below and fill in the answer boxes to complete your choice. (Type exact answers.) O A. dy OB dx The area of the given region is square units. (Type an exact answer.) 68: Table of Integrals (page 1) Substitution Rule g (b) (19(xng'(x ) ax = [ru ) au ( u =g( x ) a g (a ) Integration by Parts Juav=uv- Svau fur ax= uvid- fvu' ox Basic Integrals https://xlitemprod.pearsoncmg.com/api/v1/print/highered 51/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen | x" dx = 1x+1 + c;n# -1 On = In |x| + C cos ax dx = - si sin ax dx = - - cos ax + C tan x dx = In |sec x| + C cot x dx = In |sin> secx dx = In |sec x + tan x| + C csc x dx = - In |csc x + cotx| + C = Leax + c bax dx = = 1 a In b "bax + C; b > 0, b#1 Inx dx = x In x - x + C log bx dx = 1 Inb ( X dx = sin - 1X + C dx 1 a = -tan -1- +c dx - se a a x x2 -a2 |sin - 1x dx = xsin -1x + $1-x2 + c cos - 'x dx = xcos - 1 x - 1 - x + c tan 'x dx = x tan |sec - 1xax=xsec -1x- In (x+Vx2 -1) + c sinh x dx = cosh x + C cosh x dx = sinh > sech x dx = tanh x + C csch x dx = - coth x + C sech x tanh x dx = csch x coth x dx = - csch x + C tanh x dx = In cosh x + C coth x dx = In sin sech x dx = tan " sinh x + C= sin -1 tanh x + C csch x dx = In tanh 2 + c 69: Table of Integrals (page 2) Trigonometric Integrals cos 2 x dx = * + Sin 2x + C sin 2x dx = = - sin 2x 4 4 + C |sec ? ax dx = . csc 2ax dx = - =cotax + C tan 'x dx = tan x - x + C |cot 2 x dx = - cos 3 x dx = - - sin 'x + sin x + c sin 3 x dx = = cos 'x - cos x +C sec 3 x dx => csc 'x dx = - - csexcotx-> In /csex+ cotx| +c tan 3 x dx = > tan 2x - In /secx| + c |cot 3 x dx = - sec " ax tan ax dx = na sec "ax + C;n# 0 csc " ax cot ax dx = - - csc "ax + C;n#0 dx 1 + sin ax dx 1 - sin ax atan * + 2 dx dx 1 + cos ax - tan + C 1 - cos ax sin mx cos nx dx = - cos (m + n)x cos (m - n)x 2(m + n) 2(m - n) + C; m2 # n2 sin mx sinnx dx =- sin (m - n)x sin (m + n)x - + c; m #n2 2(m - n) 2(m + n) cosmx cos nx dx = - sin (m - n)x sin (m + n)x 2(m - n) 2(m + n) + C ;m n2 70: Table of Integrals (page 3) Reduction Formulas for Trigonometric Functions https://xlitemprod. pearsoncmg.com/api/v1/print/highered 52/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen cos "x dx = ~ cos "- 'xsin x + "-'cos -2x dx sin " X dx = - - sin "- 1x cos x + ni tan "'x dx = tan "-1x n - 1 Stan "-2 x dx; n#1 cot "x dx = _ cot" - 1x n - 1 cot n - : sec "x x= sec "-2xtanx n-2 n - 1 n - 1 sec " - 2 x dx ; n # 1 csc "x dx = _ CSc"-2x cotx n - 1 sin xcos "x dy= _ sin"-'xcos "+1x m - 1 m + n m + n.J sin m-2 xcos "x dx; m# -n sin "xcos "x dy= sin m 'xcos "-1x n - 1 m +n m +nJ sin "xcos "-2x dx; m# -n a x" sin ax dx= _ x" cosax , " fxn-1 cos ax dx; a#0 J x" cos ax dy = X"sin ax a - 2 ( xn - 1 Integrals Involving a2 - x2; a > 0 [ Vaz - x2 ax = > Vaz - x2 + , sin 1- + c a2 dx In a + Va2 -x2 xva2 - x2 X + C dx Va 2 - x 2 x2 Na2 - x2 a2x + C [x2 Val - x2 dx= (2x2 - a2 ) a2 - x2 + sin 1=+ c Vaz -x2 dx = - 1 Ja2 -x2 + C- sin -1- +c x a 12 2 - x 7 dx = - - Va2 -x2 + - sin 12 + C dx -x2 2a In -8 + C 71: Table of Integrals (page 4) Integrals Involving x2 - a2; a > 0 x2 - a2 dx = > Vx -a2 - - In x + Vx - a?|+c dx Vx 2 - 2 2 = In x + 1x - a | +c dx 1x2 - a2 + C 2 2 a x [x? Jx2 - 27 ax= (2x2 - 2? ) 1x2 - 2? - 2 in/ x + 1x-2?|+ 1x - a dx = In x + 1x2 - a21 VX - a 2 + C X dx 2 _2 5 - In |x - a 2a x + a + C dx -1 In |x2 -a21 2a x 2 + C Integrals Involving a2 +x2; a> 0 https://xlitemprod.pearsoncmg.com/api/v1/print/highered 53/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen [ Vaz + x2 ox = > Va2 + x2 + 3 In (x+ a2+x2 ) + c dx Va 2 + x 2 = In (x + Va2 +x2) +c dx In a - Val +x2 xva2 + x2 al X + C dx Va2 + x2 x 2 Na 2 + x 2 + C a x [ xar +x2 dx= (a?+ 2x2 ) Na? +x2 - 2 in (x+Val +x? ) + c [val + x2 2 dx = In x + a2 + x2 1 Va+ x + C xva +x2 a2 + x2 dx = - - In (x+ Va2 +x2)+ + c ( vaz + x ] dx = Va2 + x2 - aln a + va2 +x2 x + C dx (a2 +x2) 3/2 a2 Na? +x 2 + C dx JX (2 2 +x 2 ) - In 2a2 2 + x 2 + C 72: Table of Integrals (page 5) Integrals Involving axb; a #0, b > 0 (ax + b)" dx = - (ax + b)n + 1 a(n + 1) - + C; n# -1 ( Vax + b ) " dx = 2 ( vax + b ) n + 2 n + 2 + C; n# -2 dx 2 - tan ax - b + C dx Xvax - b Vb b xvax + b - = - = In Vax + b - vb Vb Vax + b + vb + C X x b ax + b - In lax + b| + C a dx = - ax + b 273 ( (ax + b) 2 - 4b(ax +b) +2b2 in lax + bl) + c dx 2 In |ax + b x (ax + b) 62 X + c * Vax + b d* = 2 15a (3ax - 2b)(ax + b)3/2 + c X dx = - 2 ax + b 2 (ax - 2b),ax + b + C | x( ax + b) " dx = (ax + b ) " + 1 ( ax + b b n + 2 n +1 + C; n# -1, -2 dx x (ax + b ) - In X b ax + b + C Integrals with Exponential and Trigonometric Functions eax sin bx dx = e (a sin bx - bcos bx) a2 + 62 + C eax cos bx dx =- ed(a cos bx + b sin bx) + C a2+ 62 73: Table of Integrals (page 6) Integrals with Exponential and Logarithmic Functions dx x In x = In |In x| + c [ x " Inxox = X _ xn+1 n + 1 Inx - 7 + 7 + C ;n * - 1 xe* dx = xe* - e * + c xmeax ax = 2xreax _ " [x - 1 ax dx; a * 0 [ in " xax=xin " x-n In "-1xox Miscellaneous Inte https://xlitemprod.pearsoncmg.com/api/v1/print/highered 54/61\f7/4/23, 11:16 PM Homework 8.7-Anh Nguyen 14. Use reduction formulas to evaluate the integral. 9 cot * ( 51 ) at Click here to view page 1 of the Table of Integrals. " Click here to view page 2 of the Table of Integrals. Click here to view page 3 of the Table of Integrals. " Click here to view page 4 of the Table of Integrals." Click here to view page 5 of the Table of Integrals. "Click here to view page 6 of the Table of Integrals. 79 9cot *(51) at = - cot 3(51) + 9t+ 5 cot (51) +c 74: Table of Integrals for basic forms and forms involving ax+b (page 1) Basic Forms 1. k dx = kr + C (any number k) 2. xax = " + C (n#-1) 3. 4 = In/x| + C 4. edx = etc s. fadx = 1 + c (a> 0,a# 1) 6. sin xax = -cosx + C 7. cos x dx = sin x + C 8. sec' x dix = tan .x + C csc' x dx = -cotx + C 10. sec .x tan x dx = sec.x + C 11. csc x cotx dx = -csc x + C 12. tan x dx = In |sec x| + C 13. cot x dx = In (sin x| + C 14 . sinh x dx = cosh.* + C 15. cosh x dx = sinh x + C 16. = sing + C 17. 247 - atan'd + c Via asec-i # + c Val = sinh' * + C (a> 0) L = cosh 1 + C (x> > 0) Forms Involving ax + b 21. (ar + by" dx = (ax + bjutl a(n + 1) + C , n # - 1 22. max + by dx = ( ar + 5)utl [ ar + b _ 6 + C. n # - 1,- 2 a [n+2 7+1) 23 . (ar + by' dx = =inlax + b | + c 24. max + by' dx = = - binlar + b| + c 25. ( mar + by ? ax = + Inlax + bl + ar+6 + C 26. / mar + b) = $in|ax+ 27. ( Var + 6)" dix = = Var + + C. n#-2 28. [ Nar + bax = 2Vax + 6 + 6/vax 29. (@) dx = 1In Var + 6 - Vb + c Ivax + 6 + vil (6 7Vatan ar -b + C 30. ( Var+ bax =_ Var+ 6 + 9/dx + c 31. Var +6 _Vartb_ ad + C *Var + 6 75: Table of Integrals for forms involving a 2+x^2, a^2-x^2, and x^2-a^2 (page 2) https://xlitemprod.pearsoncmg.com/api/v1/print/highered 56/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen Forms Involving a2 + x2 32. 2 dx - 4tan- + C 33 . (a + x )= = 203 ( @ + 1 7 ) + 207 tan 'd + C Ax= sinh - 1 + c = In(x+ Va +.)+ c 35 . ( Vatxax = Vato + gin( x + Vatx )+ c 36. ( 8 Va + xax = # (a + 20 ) Va +x - 4 In(x + Va+x)+ c 37 . ( Va trax = Vatx - al at Va te+ c [ Na tedx = In(x + Vatx) - Vat&+ c 39.dx = -9- In - 2In ( x + Vatx ) + mati + c -d = - Lina + Va + + + c 41. Vatx a-x Forms Involving a2 - x2 12. de = Lin + +c 43. To de 4. / Vad = sing + c 45. ( Va - x d = =Va - x+ sin-'# + C 46. X Va - Fax = $ sin's -Java - 8 (9 - 213) + C 47. / Vapidx = Va - x - aln |0 + Vo -+ + C 48. / Vat - Idx =-sing- Va -&+ C 49. vadx = 2sin'd - Java - F+ c 50. = - fina + vor + c 2Va - x Forms Involving x2 - a2 $2. / V - = In|x + Vx - al + c 53. / VP - dax = =Vr - a - $1/x + VP - al+c $4. ( V x - a ) dr = ( V 8 -@ )" WAT ( V x - a )" - 2 dx, n #- 1 55. ] (VP - Q) (n - 2)a-) ( V/x2 - 2 )1-2 1 # 2 dx ( 2 - n) n+ 2 - + c, n # -2 57. ( RV P - dax = (27 - a ) VP - a - -In/x + V-al+c so. ( V& - dx = In |x + VP - al - VR - &+ c 76: Table of Integrals for trigonometric forms (page 3) https://xlitemprod. pearsoncmg.com/api/v1/print/highered 57/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen Trigonometric Forms 63. sin ax dr = - 4 cos ar + C 64. / cos ax ax = a sin ar + C 65. sin' ardx = _ sin 2ax + C 4a 66. cos' ardx = + sin zar + c 67. sin" ardx = _ sin"- arcos ax + " - 1 /sin"-2 ax dx 68. / cos" ax ax = cos"-' ar sin ax + 1 - 1 /cost-2 ax dx 69. (a) sin ax cos by dx = - cos(a + bjx cos(a - b) . a # 12 2 (a + b) 2(a - b) sin ax sin bx dir = sin(a - b)x sin(a + b)x DM + C, d b 2(a - b) 2(a + b) (c) cos ax cos bx dx = sin(a - b)x sin(a + b)x 2(a - b) 2(a + b) " + c, b2 sin ax cos ax ax = _ Cos 20 + 4a . sin" ax cos ax ax = sin" at + C, n # -1 (n + 1 )a 72. cos ax ax = a In | sin ax| + c sin ax 73. cos" ax sin ax dx = - costtax + C. n # -1 (n + 1)a+ 74. sin ax ax = - = In | cos ax| + c a(m + n) 75. sin" arcos" ardx = _ sin' ar cos" ax + 1 - 1sin"-2 ax cos" ardx, n # -m (reduces sin" ax) a(m + n) 76. sin" arcos" ardx = Sin"tax cos" " ax + m - 1 sin" ax cos"-2 ardx, m #-n (reduces cos" ax) 77. d.x b + csinar " qv/plan ! Voctan (# - ax) + c. >2 78. dx b + c sinax ave In C + bsinax + Ve- - bcosax + C, be 82. b + ccos ax dx ave In C + bcos ar + Va - b sinax + c. 12< < b + c cos ax Trigonometric Forms 77: Table of Integrals for trigonometric forms (page 4) https://xlitemprod.pearsoncmg.com/api/v1/print/highered 58/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen Trigonometric Forms 83. / 1 + cos at = atan " + c 84. 1 - cos at = - a cot " + c 85. xsin ax ax = sinax - acosar + C 86. / xcos ax ax = cos ax + a sin ar + C 87. x* sin ax ax = - = cos ax + / x- cos ax ax 88. " cos ax ax = a sinax - 1 1-I sin ax ax 89. / tan ax dx = = In |sec axl + C 90. / cot ardx = =in |sin axl + C 91. tan' ax ax = =tan ax - x + C 92. cof ardx = - &cotax - x + C 93. / tan" ardx - tan' ' ax - tank ? ardx, n # 1 94. cor" ax ax = - aun - 1) - com -2 ardx, n # 1 95. sec ax ax = = In |sec ar + tan axl + C 96. cscar dx = - 4 In |csc ax + cot axl + C 97. sect ax dx = atan ax + C 98. csco ax dx = - acotar + C a(n - 1) 99. sec" ax ax = sect-? ax tan ax + 1 - 2 / sec"-2 ardx, n # 1 100. / csch ax ax = - csc# 2 ar cot ax + 1 - 2 / esca-2 ax dx, n # 1 a(n - 1) 101. sec" ar tan ardx = Sec ar + C, n # 0 102. csc" ax cot ardx = _ cc" at + C, n # 0 78: Table of Integrals for inverse trig forms, exponential/logarithmic forms, and forms involving square root of 2ax-x^2, a>0 (page https://xlitemprod.pearsoncmg.com/api/v1/print/highered 59/617/4/23, 11:16 PM Homework 8.7-Anh Nguyen Inverse Trigonometric Forms 103. sin-' axdx = xsin lax + 4VI - ax + c 104. cos' ardx = xcos' ax - 2VI - d-x + C 105. tan ' ardx = xtan' ax - 2 In (1 + d'x ) + C 106. x sin- ardx = 1" n + [ sin lax - 7 4 1 / V1 -ax n * -1 107. ( x" cos ' ardx = 4+ = n + [ cos ' ax + 7 4 1 / x x , n # - 1 VI - or 108. " tan ' ardx = path in + Itand ax - 7 4 , got l dx Exponential and Logarithmic Forms 109. edx = be + c 110. ba dx = 1 bex alnb + C, b > 0,b # 1 In1. xedx = = (ar - 1) + C 112. red = are - fredx 113. xbox ax = Amber alnb - ainb * 1 dx, b > 0.b = 1 114. e sin by dx = 2 (a sin bx - bcos bx) + C 115. e cos by dix = 2 + be (@cos bx + bsin bx) + C 116. Inardx = xInax - x + C 117. x" (In ax)" dx = pati (In ary n + 1 7 + 1 / x"( In ary -lax , n # - 1 118. '(In ax)" dx = (In aryat m+ 1 + c , m # - 1 119. - dy = In [In ax| + C Forms Involving V2ax - xz, a > 0 -dx = sin-1 : -") +c 121. / Vzax - x3 dx = 1 - Vzax - > + $ sin1 (1 - ") +c 122. ( Vzar - 3)" dx - (x - a)( V2ax - 13)" n + 1 n+ 1 (V 2ax - x2 )"-2 dx (x - a)(V2ax - )2-" 123. (zar - x2 )" (n - 2)7 + n - 3 (n - 2x7) ( Vzax - 12 )"- dx 124. XVzax - x' dx - ( + @)(2x - 3a)V2ax - x + 9 sin-1 (1 - a ) + c 125. V2ax - & dx = Vzax - x3 + asin-1 (1 - ") + c 126. / V2ax - dx = -2 /20 - sin-1 (1 2 4) + c 127. = asin" (" - 4) - Vzax -X + C 128. /- Max-=2 20 - x + C 79: Table of Integrals for hyperbolic forms (page 6) https://xlitemprod.pearsoncmg.com/api/v1/print/highered 60/61\f7/5/23, 2:40 AM Homework 8.8-Anh Nguyen Student: An Instructor: Zeinab Rahmanabadi Date: 07/05/23 Course: 2023SU Calculus II (MATH-2414- Assignment: Homework 8.8 43640 10-7 Compute the following estimate of f ( x ) dx using y = f( x ) the graph in the figure. T(4) 6- 10 Using the Trapezoidal Rule, T(4) = 38 Type an integer or a simplified fraction.) 2. Compute the absolute and relative errors in using c to approximate x. X = IT; C = 3.17 The absolute error is 2.84 - 10-2 (Use scientific notation. Then round to two decimal places as needed. Use the multiplication symbol in the math palette as needed.) The relative error is 9.04 - 10-3 Use scientific notation. Then round to two decimal places as needed. Use the multiplication symbol in the math palette as needed.) 3. Find the indicated midpoint rule approximation to the following integral. N e" * dx using n = 8 subintervals N (Do not round until the final answer. Then round to three decimal places as needed.) . Find the indicated trapezoid approximations to