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Definite Integration

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  1. 151.If f(a + b + 1 - x) = f(x), for all x, where a and b are fixed positive real numbers, then…
  2. 152.If [I = _1^2 dx 2x^3 - 9x^2 + 12x + 4 ], then :
  3. 153.[ _x 0 _0^x t ( 10t )dt x] is equal to
  4. 154.Let a function f : [0, 5] [ ] R be continuous, f(1) = 3 and F be defined as : [F(x) = _1^x…
  5. 155.The value of [_0^2 x ^8x ^8x + ^8x dx] is equal to :
  6. 156.If for all real triplets (a, b, c), f(x) = a + bx + cx2; then [_0^1 f(x)dx ] is equal to :
  7. 157.Let P(x) = x2 + bx + c be a quadratic polynomial with real coefficients such that [_0^1 P(…
  8. 158.Let g(x) = [_0^x f(t)dt ], where f is continuous function in [ 0, 3 ] such that [1 3] [ ] …
  9. 159.Let f : R [] R be a continuous function.
  10. 160.If the real part of the complex number [(1 - + 2i )^ - 1] is [1 5] for [ (0, )], then the …
  11. 161.If [_0^100 ^2x e^( x - [ x ] )dx = ^3 1 + 4 ^2, R ] where [x] is the greatest integer less…
  12. 162.Let f be a twice differentiable function defined on R such that f(0) = 1, f'(0) = 2 and f'…
  13. 163.If [I_n = _ 4^ 2 ^nx,dx ], then :
  14. 164.[ _n [ 1 n + n (n + 1)^2 + n (n + 2)^2 + ........
  15. 165.Let [f:[0, ) [0, )] be defined as [f(x) = _0^x [y]dy ] where [x] is the greatest integer l…
  16. 166.The value of [_ - 2^ 2 ( 1 + ^2x 1 + ^ x ) ,dx] is
  17. 167.The value of [ _n 1 n_r = 0^2n - 1 n^2 n^2 + 4r^2 ] is :
  18. 168.Let [f(x) = _0^x e^tf(t)dt + e^x ] be a differentiable function for all x[]R.
  19. 169.For x > 0, if [f(x) = _1^x _et (1 + t)dt ], then [f(e) + f( 1 e )] is equal to :
  20. 170.The value of [_ - /2^ /2 ^2x 1 + 3^x dx] is :
  21. 171.If [U_n = ( 1 + 1 n^2 )( 1 + 2^2 n^2 )^2.....( 1 + n^2 n^2 )^n], then [ _n (U_n)^ - 4 n^2]…
  22. 172.The value of the definite integral [_ - 4^ 4 dx (1 + e^x x)( ^4x + ^4x) ] is equal to :
  23. 173.If [x] is the greatest integer [] x, then [ ^2_0^2 ( x 2 )(x - [x] )^[x]dx] is equal to :
  24. 174.Let f be a non-negative function in [0, 1] and twice differentiable in (0, 1).
  25. 175.The value of the integral [_ - /2^ /2 dx (1 + e^x)( ^6x + ^6x) ] is equal to
  26. 176.[ _n ( n^2 (n^2 + 1)(n + 1) + n^2 (n^2 + 4)(n + 2) + n^2 (n^2 + 9)(n + 3) + ,,....,, + ,,n…
  27. 177.[ _n 1 2^n( 1 1 - 1 2^n + 1 1 - 2 2^n + 1 1 - 3 2^n + ,,...,, + ,,1 1 - 2^n - 1 2^n )] is …
  28. 178.Let [[t]] denote the greatest integer less than or equal to t.
  29. 179.For any real number x, let [[x]] denote the largest integer less than equal to x.
  30. 180.If [b_n = _0^ 2 ^2nx xdx,,n N ], then
  31. 181.The value of [_0^ e^ x x (1 + ^2x)(e^ x + e^ - x)dx ] is equal to:
  32. 182.[ _0^20 (| x|+| x|)^2 d x is equal to ]
  33. 183.Let [f(x)=2+|x|-|x-1|+|x+1|, x R].
  34. 184.[_0^2(|2 x^2-3 x|+[x-(1)/(2)]) d x], where [t] is the greatest integer function, is equal …
  35. 185.Let [f: R R] be a function defined as [f(x)=a (([x])/(2))+[2-x], a R] where [[t]] is the g…
  36. 186.Let [ I=_ / 4^ / 3((8 x- 2 x)/(x)) d x ].
  37. 187.If m and n respectively are the number of local maximum and local minimum points of the fu…
  38. 188.Let f be a differentiable function in [( 0, 2 )].
  39. 189.Let [I_n(x)=_0^x 1(t^2+5)^n d t, n=1,2,3, .] Then :
  40. 190.The minimum value of the twice differentiable function [f(x)=_0^x e^x-t f^(t) dt-(x^2-x+1)…
  41. 191.Let [t] denote the greatest integer less than or equal to t.
  42. 192.If [[t]] denotes the greatest integer [ t], then the value of [_0^1[2 x-|3 x^2-5 x+2|+1] d…
  43. 193.The integral [_0^()/(2) (1)/(3+2 x+ x) ~d x] is equal to :
  44. 194.If [_0^2 ( 2x - 2x - x^2 )dx = _0^1 ( 1 - 1 - y^2 - y^2 2 )dy + _1^2 ( 2 - y^2 2 )dy + I ]…
  45. 195.[ _n _r = 1^n r 2r^2 - 7rn + 6n^2 ] is equal to :
  46. 196.Let f be a continuous function satisfying [_0^t^2(f(x)+x^2) d x=(4)/(3) t^3, t > 0].
  47. 197.If [f: R R] be a continuous function satisfying [_0^()/(2) f( 2 x) x d x+ _0^()/(4) f( 2 x…
  48. 198.Let the function [f:[0,2] R] be defined as [f(x)= casese^ x^2, x-[x], & x [0,1) e^[x- _e x…
  49. 199.The value of the integral [_- _e 2^ _e 2 e^x( _e(e^x+√1+e^2 x)) d x] is equal to :
  50. 200.Among (S1): [_n 1n^2(2+4+6+ +2 n)=1] (S2) : [_n 1n^16(1^15+2^15+3^15+ +n^15)=(1)/(16)]