INTEGRALS tan C i o l u t i o n Example30 (x) dr =0 iffis an -a function. Solution Odd. 2a Fample31 Jf() dr = 2Jf(x) dr, iff(2a-)= Solution (r). Example 32Sin sin" x dx x+cos" x Solution 4 7.3 EXERCISE Short Answer (S.A.) Verify the following: 1. d =x-log (2x+31+C 2x+3 2. xSdr = +3x log k? +3x| +C Evaluate the followingg 6log 3. 2)d x+1 J 4logr hgr 163 164 MATHEMATICS 6. l+cosx +Cosx x+SinxX Sin x+COS x 1 tan' xsec' xdr 9. V+sin xdr 8. yx =2) 10. (Hint: Put 12. (Hint: Put x = z) +sin 2x=dx a+x 11. a-x 13. 1+r dt 14. 16-92 15. 3-2 16. 4 17. V5-2x+x'd 18. 19. Jds putr-t 20. 22 V2ax-xdr 21. (cos Sx +cos 4x) dx 1-2cos3x sin dx (1-) sin° x +cos * 23. dr sin x cos' x INTEGRALS 25. dr COs x- cos 2x 1-cosx - 165 dx Hint: (Hint : Put x = sec 0) lhuate the following as limit of sums: J+3)dr 28. Je d Evaluate the following: dx 30. e+e tan xdr 1+m tan' x dr 31. xdx ir-1)(2-x) 32 dr 33 x sin xcos 34. xd 0 1+r)1-x (Hint:letr= sin6) Long Answer (LA.) rdr 35 37. xdx -12 36. 2x-1 38. 1+sin x (r+ a Xx+b') dx (x-1)(r+2)(r-3) 166 MATHEMATICS n' 40. +rtx Vatx 39. 42. +cos.x* (Hint: Put Je cos'x d x = a tan') 5 41. (-cosx)2 43. 44 tan xdr (Hint: Put tanx = P) dx (a cos x+b° sin* x) (Hint: Divide Numerator and Denominator by costx) 45. 47. 46. r log(1+2x)dr xlogsin xdr log(sin x+cosx)dx Objective Type Questions Choose the correct option from 48. given four options in each of the Exercises from 48 to 0 cos2x-cOS 4D dr is equal to COS X-Ccos (A) 2(sinx +xcos8) +C (C) 2(sinx + 2xcos6) +C (B) 2(sinx - xcos6) +C (D) 2(sinr -2x cose) +C INTEGRALS 167 dx is equal cqual to to * sinr-a) Sin(x-b) (A) sin ( -a) log Isin(x-b+C sin(x-a) C)cosec (b-a) log . 4 sinx-0)+C |sin(x - a) (B) cosec (b - a) log (D) sin (b- a) log sin(x-a)+C sin(x b) sinx-a)c sin(x - b) tan x dr is equal to (A)(r+1) tan" vx-vx+C (B) xtan-V+C (C) x-x tan" vx +C D) - x +1)tan" vx +C dx is equalto (B) e (A) 4+C C (1+ 52. (4x+1 (D) (1+ +C dr is C equalto +C C)(1+4)*+C 10x 168 MATHEMATICS dx 53. If g+2(r? +1) log|l +r|+b ? Ca1054. (B) a= 0b- D)ab 2 is equal to0 54. x -loglh-d+ C (C)x -logl1+ xl+C (A) 56. x+21+C, then r4 Cx+2+14 (A)a10,b 5 55. tan'x +log (B) x+ -logl1-xl+C (D) x+logl1+a+C (X+Sin x a is equal to 1+cosx (A) logll+ coszl+ C (B) loglx+ sin x+ C (C)x-tan+C D) x.tan +C x dx If VI+ al+*)i+by1+x+C,then (A) a b=1 C)a b=-1 (B)a (D)a- b=1 b b-1 INTEGRALS d 169 is equal to +cos2 (B) 2 (A)1 C) 3 (D) 4 C)2 D) 22-1) -sin 2xdx is equal to 0 B) 2 (2+1) (A) 22 59. cosxe"dr is equal to 0 fI+3 60. Fill in the blanks in each of the following Exercise 61. 62. 63. dx = 1+4x Sin x then 8 a = -dx 3+4cos x The value ofsin'x cosx dx is 60 to 63. ANSWERS 295 33. Rs 1920 36. B 37. A 40. A D 41. A 44. B B A B .(1, o) 42. D 45. C 48. A D 38. C 46. B 49. B 52. C 50. C 53. B 54. C 56. A 57. B 58. B 60. (3,34) 61. x+y =0 62. (-oa-1) 64. 2ab 7.3 EXERCISE 4tC 4. -x+3logr+1+c 6 tanx tan tan+C 7. 2 5 +C 8. x+c 3 9-2cos+2Sin*C 10. 11.-acos 12 3 4-log|l+|+ 14. sin +c 15. i 5. log|x +sin +c .* 16. 3W+9-loglr+ + 9+ 3x 296 MATHEMATICS 17. 2 s-2r+r +2log/x-I+s-2x+r|+e 18.log|-bgls' +e 4 20. 2ar- 22. sin 2x +sin" sin x + . lo23. tanx c - logm 28. -1 30. m21 32. 2-1 33. 34. X a tan-btan 40. 3x + c 31. 7T 3 35. o log - 27. sec ()+e 2 29. tan 'e 38. cot x 25. 2 sinx +x+c 24.in 26. tan' x+c 19. -3 x-16 (x +2) +c c 37. T 39. xean +C sunbs sI '¬7 v 97 spun b s siIunbs g "T7 '91 81 syunbs p bs 'SI 'II 07 LI SI s u n"bs sun'bs 6 Tsun '8 91 sun'bs 9 1 SI saun b s 61 sunbs ol+E sun bs , sun'bs syunbs 'EI bI 6 sun'bs 96 0 suunbs 91 n bs sunbs uz spuab s t 9 S sunbs saunbs sun bso1 E sun bs sun "bs ISIONTxIE8 o9 19 z9 + v I-a '6S 09 34 9s LS V8t v 0S IS 6 309 1 I ++uez+xuej| NTXuei7 I+X ueizA-x ue I - Xuej +xs0g-xus L67 ) +|xgs0o-N{ us S8IMSNY