Section 8.4 Reading Assignment: Trigonometric Substitution

Answer Only Exercise 1, 2, and 3 by using a screenshot provided Calculus Pearson textbook. Make sure you read these three questions very carefully and see on what it is asking for.

References: Thomas' Calculus: Early Transcendentals | Calculus | Calculus | Mathematics | Store | Pearson+

Section 8.4 Reading Assignment: Trigonometric Substitution Instructions. Read through this assignment and complete the three exercises below by reading the appropriate passages of the textbook. The basic premise of trigonometric substitution is backwards from the standard usubstitution. When looking at the usubstitution formula: f f(9(x})9'{x) dx = [ fEu) du We start with the lefthand side and setting or = gfx}, do = g'i'x) dx, results in the rightha nd side. In trigonometric substitution, we actually go the other way arOund: [ fix) dx = f tomato am We start with the lefthand side and setting x = 9(9), dx = g'iEl] dB, to get a messier righthand side.jl This is usually counterproductive, which is why we have never gone that route before. But we can use trigonometric identities to turn certain integrals into trigonometric integrals, which we learned how to deal with in Section 8.3. Exercise 1. Read from the start of the section to the start of Example 1 (p. 480 481). Explain what reference triangles are and how they can be used to determine which trigonometric substitution to use. These reference triangles come in handy to know which substitution to use along with some trigonometric formulas using right triangle trigonometry (Le. SOH CAH TOA]. Pay close attention to these examples and be sure to understand the process. In particular, it's quite important to remember the differentials (dx) also Substitute. This is probably the most often forgotten part of this process. Exercise 2. Read the boxed text "Procedure for a Trigonometric Substitution" {p. 481} and Example 1 (p. 481 482). Explain how the first two steps ofthe procedure are clone in this example. Include the formulas here. Exercise 3. Read Example 1 (p. 481 482) and look at Figure 8.4 (p. 482). Explain how the reference triangle in the figure and right triangle trigonometry are used to convert the answer from an expression in E! to an expression in x in the last equality ofthe computation in Example 1. One thing to note in Example 1 (and also in Examples 3 and 4] is the absolute value in the integral after doing the trigonometric substitution. In Example 1, this appears as |sec(B)| on the top of p. 482. This is a technical detail that we will avoid by working where the functions in question are positive. 1 While theta [B] is often used in trigonometric Substitution to highlight the connection with angles, it's often useful to use a different letter like t instead, especially when typing since 9 is a bit harder to type in general. Chapter 8 Techniques of Integration 480 Chapter 8 Techniques of Integration Assorted Integrations 41. / secto do 42. / 3 sect 3x dx Use any method to evaluate the integrals in Exercises 63-68. 63. 64. sin'dx 43. / cso' 9 do 44. / sech x dx tan x cos'x 65. tan' 66. 45. / 4 tan x dx CSC TOx col & x 6 tant x dx cos X 47. tan' x dx 48. / cone Zr dx 67. / xsin' x di Applications 49. cot' x dx 50. / 8 cott dt 69. Are length Find the length of the curve Products of Sines and Cosines y = In (sin x). Evaluate the integrals in Exercises 51-56. 70. Center of gravity Find the center of gravity of the region 51. / sin 3.x cos 2x dx 52. / sin 2x cos 3x dx bounded by the x-axis, the curve y = sec x, and the lines x- -1/4, x = 1/4. 53. sin 3x sin 3x dx 54. 71. Volume Find the volume generated by revolving one arch of the sin x cos x x curve y = sin x about the x-axis. 72. Area Find the arca between the x-axis and the curve y = 55. / cos 3x cos 4x x 56. cos x cos 7x dx VI + cos 4x, 0 5 X 5 5. 73. Centroid Find the centroid of the region bounded by the graphs of y = x + cos x and y = 0 for 0 s x s 2v. Exercises 57-62 require the use of various trigonometric identities before you evaluate the integrals. 74. Volume Find the volume of the solid formed by revolving the re- gion bounded by the graphs of y = sinx + sec x. y = 0, x = 0, 57. sin' # cos 30 de 58. cos' 20 sin A d# and r = w/3 about the x-axis. 75. Volume Find the volume of the solid formed by revolving the region bounded by the graphs of y = tan" 's, x = 0, and y = 1/4 59. cos # sin 20 9 60. sin A cos 20 de about the y-axis. 76. Average Value Find the average value of the function 61. sin e cos A cos 38 de 62. sin # sin 24 sin 39 dip 1 - sin d A on [ 0. #/6].8.4 Trigonometric Substitutions Trigonometric substitutions occur when we replace the variable of integration by a trigo- nometric function. The most common substitutions are a = a tan 0, x - a sind, and x = a sec 8. These substitutions are effective in transforming integrals involving Vatx. Va' - x, and Vx - a into integrals we can evaluate directly since they come from the reference right triangles in Figure 8.2. Va-+ 12 Val - r= asin A Val tx- asco Va' - - acoso) Vx - a' - atan of FIGURE 8.2 Reference triangles for the three basic substitutions identi- fying the sides labeled x and a for each substitution.Chapter 8 Techniques of Integration 8,4 Trigonometric Substitutions 481 With x = a tan d, atx= at a tan'd = all + ung = a sec'e. With x = a sin d, D = tan al - x = al - a sino = a(1 - sin? #) = a cos28. With x = a sec d, x - a = a sec3 0 - al = a (sec] 0 - 1) = a tan' 0. We want any substitution we use in an integration to be reversible so that we can change back to the original variable afterward. For example, if x = a tan 0, we want to be T able to set o = tan" (x/a) after the integration takes place. If x = a sin 0, we want to be able to set # = sin ' (x/a) when we're done, and similarly for & = a sec . As we know from Section 1.6, the functions in these substitutions have inverses only for selected values of # (Figure 8.3), For reversibility, r = a tan 0 requires 8 = tan ! with 2 x = a sine requires e = sin with W- 121, if a x = a seco requires 0 = sec with 0.Procedure for a Trigonometric Substitution 1. Write down the substitution for x, calculate the differential dx, and specify the selected values of # for the substitution. 2. Substitute the trigonometric expression and the calculated differential into the integrand, and then simplify the results algebraically. 3. Integrate the trigonometric integral, keeping in mind the restrictions on the angle # for reversibility. 4. Draw an appropriate reference triangle to reverse the substitution in the inte- gration result and convert it back to the original variable r. EXAMPLE 1 Evaluate 14+ x Solution We set x = 2 tan e. dx = 2 sec- 8 do. 2 4 + x' = 4 + 4 tan 0 = 4(1 + tan' 0) = 4 sec20.Chapter 8 Techniques of Integration 482 Chapter 8 Techniques of Integration Then V4+ dx V4+ = 2 sec' 8 de = sec- 9 de Vice- A = |sec el V4 sec- d |sec el 2 FIGURE 8.4 Reference triangle for sec 9 do seco > 0for - 7 0 67 * = 2 tan # (Example 1): = In sec o + tan 0 + c = In and 2 V4+ + * + C From Fig. 8.4 seco - V4+.x 2 Notice how we expressed In sec # + tan # in terms of x: We drew a reference triangle for the original substitution x = 2 tan 0 (Figure 8.4) and read the ratios from the triangle. EXAMPLE 2 Here we find an expression for the inverse hyperbolic sine function in terms of the natural logarithm. Following the same procedure as in Example 1. we find that d.x Vat = sec 0 de x = atan e, de = a sec A de = In |sec # + tan #| + C = In Fig. 8.2 From Table 7.9, sinh '(x/a) is also an antiderivative of 1/ Va' + x', so the two anti- derivatives differ by a constant, giving sinh-14 = In + C.Setting x = 0 in this last equation, we find 0 = In |1| + C, so C =0. Since Val + x > x , we conclude that sinh -1 4 Vatr = In + (See also Exercise 76 in Section 7.3.) EXAMPLE 3 Evaluate x3 d.x 19- x Solution We set x = 3 sind, dx = 3 cos 0 de. 2 9 - x = 9 - 9sin 0 = 9(1 - sin 0) = 9 cos' d