2. Compute the indefinite integral 4 fe2x In(er + 1) dr by using a convenient substitution. [5] A. (e* + 1)? In(e" + 1) - 4e" + C, where C is an arbitrary constant. B. ve2x + 1 In(e# + 1) +ex - e2x + C, where C is an arbitrary constant. C. (e2x - 1) In(er + 1) + eax + C, where C is an arbitrary constant. D. 2(e2x - 1) In(e* + 1) - e"(e - 2) + C, where C is an arbitrary constant. E. vezx + 1 In(e* + 1) - 2e" + C, where C is an arbitrary constant. F. (ezz - 2) In(e* + 1) - ex + C, where C is an arbitrary constant. G. 4(ex + 1)3 In(e# + 1) - e" + C, where C is an arbitrary constant. 3. Evaluate the indefinite integral 2 fr tan -1(x) dr using integration by parts. [5] A. x' tan-1(x) - In(1 + z?) + A, where A is an arbitrary constant. B. 2rcot-1(x) - In(1 + z?) + A, where A is an arbitrary constant. C. 2rtan-1(x) - In(1 + x2) + A, where A is an arbitrary constant. D. 2x2 cot(x) - In(1 + x?) + A, where A is an arbitrary constant. E. (x2 + 1) tan (x) - 2 + A, where A is an arbitrary constant. F. xtan(x) - x? In(1 + x?) + A, where A is an arbitrary constant. G. (x2 + 4) cot-1(x) - In(1 + x2) + A, where A is an arbitrary constant. 4. Compute the indefinite integral Jendra by employing a partial fraction decomposition of the inte- [5] grand A. 3x + + C, where C is an arbitrary constant. B. 3x + In ( fajr + C, where C is an arbitrary constant. C. In 2-D + x2 In|x - 1/C, where C is an arbitrary constant. D. 3In #+2 + C, where C is an arbitrary constant. E. 3x2 + + C, where C is an arbitrary constant. F. 3 + In + + C, where C is an arbitrary constant. G. 3r + $3 + C, where C is an arbitrary constant. 5. Evaluate the indefinite integral 8 f x2 v1 - x2 da by using a convenient trigonometric substitution. [5] A. xv1 - x2 - 4cot-1(x) + 2x2 + K, where K is an arbitrary constant. B. x'v1 - x-(x2 -1) - 4cot-1(x) + K, where K is an arbitrary constant. C. xV1 - 2' - 4sec-1(x) + K, where K is an arbitrary constant. D. xV1 - x2 - 4cos-1(2) + K, where K is an arbitrary constant. E. xv1 - x2(2x2 - 1) + sin 1(x) + K, where K is an arbitrary constant. F. 4xv1 - 12 - 4 tan-1(2) + K, where K is an arbitrary constant. G. 8v1 - 12 - 4(2x2 - 1) tan '(2) + K, where K is an arbitrary constant. 6. Solve the separable ODE [5] dy y2 + 3y +2 A. y = Me#2, where K is an arbitrary constant. B. y = Met, where K is an arbitrary constant. C. 2y3 + 9y' + 12y = 3x' + K, where K is an arbitrary constant. D. y = In (142 2 ), where K is an arbitrary constant. E. y = Ke-2, where K is an arbitrary constant. F. y = In (141 ), where K is an arbitrary constant. G. 1=7. Solve the separable ODE dx dy = xy + 2y+ +2. A. In ly + 2) = 2/2 + x. B. In |x + 2) = y?/2 + y + C, where C is an arbitrary constant. C. In ly + 21 = x2/2 + a + C, where C is an arbitrary constant. D. In|x - 2) = y?/2 + y + C, where C is an arbitrary constant E. In ly + 2| = x2/2 - x. F. In )x + 2) = y?/2 - y + C. where C is an arbitrary constant G. Inly - 21 = 12/2 - x + C, where C is an arbitrary constant 8. The separated form of the separable ODE a(x) 24 - b(x)y = 0 is A. ydy = -a(x)b(x)dr. _LO dr. B. = -80 "Odr. _ dr. du = -60 E. du = LO dr. F. = - 2 _LO de. a(2) LO dr. G. ydy = -1) 9. Solve the homogeneous first-order ODE. dy 2 + da tan(y/x) A. In(cos(y/x)) - In |x| = K, where K is an arbitrary constant. B. sin '(y/x) - In|x) = K, where K is an arbitrary constant. C. In | cos(y/x)| + In |x| = K, where K is an arbitrary constant. D. cos '(y/x) - In |x) = K, where K is an arbitrary constant. E. sin (y/x) - In |x] = K, where K is an arbitrary constant. F. cos '(y/x) - In |x| = K, where K is an arbitrary constant. G. sin(y/x) - In |x] = K, where K is an arbitrary constant. 10. Solve the ODE dy - 2x a + 2y A. In(vx2 + y?) + tan-1(x/y) = K, where K is an arbitrary constant. B. In((x2 + y?) - tan(y/x) = K, where K is an arbitrary constant. C. In(Vx' + y ) + tan '(y/x) = K, where K is an arbitrary constant. D. In(vx2 + y?) - tan '(y/x) = K, where K is an arbitrary constant. E. In(x2 + y?) + tan '(y/x) = K, where K is an arbitrary constant. F. In(\\x2 + y?) + tan(y/x) = K, where K is an arbitrary constant. G. In(x2 + y?) - tan '(y/x) = K, where K is an arbitrary constant