Indefinite Integration PDF

INDEFINITE INTEGRATION 1. If f(x) and g(x) are two 21. If x > 1, then functions such that f(x) = g(x) then f(x) is called antiderivative of g(x) with respect to x. 2. If f(x) is an antiderivative of g(x) then f(x)...

INDEFINITE INTEGRATION 1. If f(x) and g(x) are two 21. If x > 1, then functions such that f(x) = g(x) then f(x) is called antiderivative of g(x) with respect to x. 2. If f(x) is an antiderivative of g(x) then f(x) + c is also an antiderivative of g(x) for all c  R. – cosec–1x + c and if x < –1 then 3. If F(x) is an antiderivative of f(x) then F(x) + c, cR is called indefinite integral of f(x) with respect to =. x. It is denoted by dx. The real number c 22. is called constant of integration. 23. 4. The integral of a function need not exists. If a 24. function f(x) has integral then f(x) is called an integrable function. 25. 5. The process of finding the integral of a function is 26. known as Integration. 27. 6. The integration is the reverse process of differentiation. 28. 7. If n  –1, then +c. 29. 8. 30. If f(x) is an integrable function and k is a real number then. 9. +c. 31. f(x), g(x) are two integrable functions then. 10. 32. If f(x), g(x) are two integrable functions then 11.. 33. If are integrable 12. a > 0, a  1, then functions then = 13.. 34. If f(x), g(x) are two inegrable functions and k, l are 14. two real numbers then 15. 16. 35. If then 17.. 18. 36. If f(x) is a differentiable function then = log |f(x)| + c. 19. 37. dx = log |sec x| + c. 20. 38. dx = log |sin x| + c. 39. dx=log |secx+tanx|+c = log |tan(4+x/2)| +c 40. dx = log |cosecx – cot x| + c = log |tan x/2| + c 229  Indefinite Integration 41. If f(x) is differentiable function and n  –1, then 55. If the given integral is of the form then take px + q =. 42. If n = –1, then. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. If the given integral is of the form then take px + q =. 54. If the given integral is of the form then take px + q =. 230  Indefinite Integration 56. If the given integral is of the form 63. If u and v are two functions of x then. , then put px + q = 64. If u and v are two functions of x; u, u, u … denote the successive derivatives of u and v 1, v2,. v3, … denote the successive integrals of v then the extension of Integration by parts is = 57. If the given integral is of the form uv1 – uv2 + u v3 – uv4+……. dx then put x =. 65. In integration by parts, the first function will be taken as in the following order. Inverse functions, logarithmic functions, Algebraic functions, 58. If the given integral is of the form Trigonometric functions and exponential functions. (To remember this a phrase ILATE). or dx or or 66. (a cos bx + b sin bx) dx then put ax + b = t2 and + c. hence dx = 2t dt. 67. (a sin bx – b cos bx) 59. If the integral is of the form 68.. or 69. 70. If In = then then , where n is multiply both numerator and denominator with sec2x and take tan x = t. the +ve integer. 71. If In = then 60. If the integral is of the form or. or , take 72. If In = then =t  = dt .  73. If In = then (1 + t2) dx = 2 dt  dx =.. Sin x = , 74. If In = then cos x = =.. 61. If the integral is of the form 75. If In = then , take acos x + b sin x =. A (c cos x + d sin x) + B(c cos x + d sin x). 76. If In = then 62. Integration by Parts : If f(x) and g(x) are two. integrable functions then 77. If , then. 231  Indefinite Integration EXERCISE – I 1. dx = 78. , where t = 1) 2) ax + b. 3) 4) 79. 2. 80.. 1) 2) 81. where t = xn. 3) 4) 82.. 3. dx = 83. If a > b then 1) 2) – 3) 4). 4. dx= 84. 1) 6x + 2 log |x| – +c 2) 6x + 5 log |x| + +c 85. 3) 6x – 4 log |x| + +c 4) 2x – 5 log |x| – +c 86.. 87. [a sin (bx + c) – 5. dx = b cos (bx + c)] + k. 1) 2) 88. [a cos (bx + c) 3) 4) + b sin (bx + c)] + k. 6. dx = 89. [a sin (bx + c) – 1) b cos (bx + c)] – – 2). *3) 90. [a cos (bx + c) + 4) b sin (bx + c)] – 7. dx = + 1) 2) 3) 4). 8. dx = 232  Indefinite Integration 1) cosx +c 2) sinx + c 3) tanx + c 4) cotx + c 21. dx = 9. dx = 1) sin x + cos x + c 2) tan x + cot x + c 3) sec x – cosec x + c 4) sin x – cos x + c 1) x + c 2) 3x2 +c 3) +c 4) 22. dx = 10. dx = 1) – 3 cot x + 2 tan x + c 2) 3 cot x + 4 tan x + c 3) –3 cot x + 4 tan x + c 4) –3 cot x – 4 tan x + c 1) ex + c 2) x + c 3) 2ex + c 4) 2x2 + c 11. dx = 23. = 1) 5x + c 2) 3) log5+c 1) log cos hx + c 2) log(cos hx+sin–1 hx)+c 4) x 5x + c 3) +c 4) x + c 12. dx = 1) – cot x – x + c 2) tan x – x + c 24. dx = 3) cot x – x + c 4) sin x – x + c 1) sec hx + c 2) –sec hx + c 13. dx = 3) tan hx + c 4) – tan hx + c 1) – cot x – x + c 2) tan x – x + c 25. dx when 3) cot x – x + c 4) sin x – x + c 0