Question 12

3/81 / x / x− / 13. (1 ) (2 ) x x x e e e+ + 14. 2 2 ( 1) ( 4)x x+ + 15. cos 3x elog sin x 16. e3 logx (x4 + 1) - 1 17. f ′ (ax + b) [f (ax + b)]n 18. 3 sin sin ( )x x + α 19. 1 x x − + 20. 2 sin 2 1 cos 2 xx ex + + 21. 2/2 ( 1) ( 2) / x x / x x + + + + 22. - 1 1tan 1 x x − + 23. 2 2 1 log ( 1) 2 logx x x x  + + −  Evaluate the definite integrals in Exercises 24 to 31. 24. 1 sin 1 co s π π −   − ∫ x xe dxx 25. 4 4 40 sin cos cos s in x x dx x x π +∫ 26. 2/2 2 20 cos cos 4 sin x dx x x π +∫ 27. 3 sin cos sin 2 x x dx x π π + ∫ 28. 0 1 dx x x+ −∫ 29. 4 sin cos 9 16 s in 2 x x dxx π + +∫ 30. 12 sin 2 tan (sin )x x dx π − ∫ 31. 4/1 [| 1| | 2 | | 3|]x x x dx− + − + −∫ Prove the following (Exercises 32 to 37) 32. 3/21 2 2 log 3 3( 1) dx x x = + +∫ 33. 1/0 1xx e dx =∫ 34. 1 17 4 cos 0x x dx − =∫ 35. 32 2sin 3x dx π =∫ 36. 34 2 tan 1 log 2x dx π = −∫ 37. 1 1 sin 12x dx− π= −∫ Choose the correct answers in Exercises 38 to 40 38. x x dx e e −+∫ is equal to

(A) tan-1 (ex) + C (B) tan-1 (e-x ) + C (C) log (ex - e- x) + C (D) log (ex + e- x) + C 39. 2 cos 2 (sin / cos ) x dx x x+∫ is equal to (A) -1 Csin cosx x ++ (B) log |sin cos | Cx x+ + (C) log |sin cos | Cx x− + (D) 2

(sin cos )+x x 40. If f (a + b - x) = f (x), then ( ) b a x f x dx∫ is equal to (A) ( )2 / b / a a b f b x dx+ −∫ (B) ( )2 b a a b f b x dx+ +∫ (C) ( )2 / b / a b a f x dx− ∫ (D) ( )2 b a a b f x dx+ ∫