x2 - 8, 281 Evaluate:: dr 14 (i) What is the correct trigonometric substitution to find an antiderivative: O = = 91 sec(0) O x = 91 sin(@) O z = 91 tan(@) O r = 8, 281 tan(0) O x = 8, 281 sec(@) (ii) Which one is the correct statement after substituting the above trigonometric function: 0 VIZ - 8,281 de = 1 14 912. sin2(0) cos(@) de O VI2 - 8, 281 14 dz = 91 / sin2(@) cos(@)de O Vz2 - 8, 281 de = 1 912 sec2 (0) tan(0) de 24 O Vz2 - 8, 281 de = sin2(0) cos(0) de 12 - 8, 281 (iii) What is the most general antiderivative of VE 14 Answer: Vz2 - 8, 281 de = 241 Evaluate:: 12 9, 604 - 12 (i) What is the correct trigonometric substitution to find an antiderivative: Or = 98 sin(@) Or = 98 sec(0) tan(@) O x = 9, 604 sec(@) O I = 98 sec(@) Or = 98 tan(@) (ii) Which one is the correct statement after substitting the above trigonometric function 1 o J 1 sec- (0) de 12 /9, 604 - 12 982 1 di = csc 2 (@) de 121/9, 604 - 12 982 1 de = 1 98 tan?(0)de 121/9,604 - 12 1 de = 1 982 cos (8 ) de 121/9, 604 - 12 1 (iii) What is the most general antiderivative of 12 9, 604 - 12 Answer: 1 di = 12 19,604 - 121 Evaluate: V/7,396 + 12 (i) What is the correct trigonometric substitution to find an antiderivative: Of = 7, 396 tan(@) O = = 86 sec(@) O x = 7, 396 sec(@) O x = 86 tan(@) O r = 86 sin(@) (ii) Which one is the correct statement after substituting the above trigonometric function: 1 - de = sec(0)de V/7,396 + 12 1 de = 86 cos(0)de V/7, 396 + 22 1 O . = / cos(@)de V/7, 396 + 12 1 O dz = 86 /tan?(0)de V7, 396 + 2 2 1 (iii) What is the most general antiderivative of V7, 396 + 12 Answer: 1 de = V7, 396 + 12Let us consider the following rational function: f(z) = 41x +15r+16 13-12+1-1 (i) What is the corrct form to find the partial fractions of f (I) O 41x-+15r+16 A + BitC 13-12+1-1 I-1 4lr +15r+16 A Br+C + 13-12+1-1 12+1 412+151+16 A + BI 13-12+1-1 I-1 12+1 41x2 +15r+16 C + 23-12+1-1 12 +1 (ii) Write the partial decomposition of this rational function: Answer. 41x2+15r+16 13-12+1-1Let us consider the following rational function: 4,747r+1,720 f(I) = 6,51012 +3,730r+400 Answer. (i) Write the partial decomposition of this rational function 4,747r+1,720 6,510r2+3,730r+400 (ii) Evaluate [ f(x)dx Answer. 4, 747r + 1, 720 de = 6, 510r2 + 3, 730x + 400