I want ans asap

cos 3x sin c 31. 32 (a2 + 1) 2 dx dx 2 + 4x + 5 " cos 2x 33. dx 34. x sin x dx 4 + 1 4 + 1 cos x 35. r sin x dx 36. (2 + 1) (x2 + 4)" dx (x2 + 1) (2 + 9) 37. sin x x ti . dx [Hint: First substitute sin x = (ex - e-2x) /2i, x real.] 8 38. cos x + x sin x x2 +1 dx [Hint: Consider eiz / (z - i).] In Problems 39-42, use an indented contour and residues to establish the Cauchy principal value of the given improper integral. 39. sin -dx = 1 40. sin x -oo 2 (x2 + 1) -dx = 1(1 -e-1) 1 - cos x x cos x 41 dx = 42 too a2 - 3x + 2 dax = [sin 1 - 2 sin 2] 6.6.3 Integration along a Branch Cut In Problems 43-46, proceed as in Example 6 to establish the Cauchy principal value for the given improper integral. 1 43. dx = 44 1 Vac (a2 + 1) V2 So Vac ( ac + 1 ) (20 + 4 ) dx = 7-1/3 45 2 7T ( 2 + 1) 2 dix = 4V2 46. ( x + 1)2 dx = 3V3 In Problems 47 and 48, establish the Cauchy principal value for the given improper integral. Use Problem 47 to verify the answer in Example 6. Use Problem 48 to verify the answer to Problem 45. ra-1 IT 47 . - dx = 0