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Definite Integration

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  1. 51.Let f,g:(0,) R be two functions defined by f(x)= _-x^x(|t|-t^2)e^-t^2dt and g(x)= _0^x^2t^…
  2. 52.Let f:R R be a function defined by f(x)=x(1+x^4)^1/4, and g(x)=f(f(f(f(x)))).
  3. 53.Let y=f(x) be a thrice differentiable function in (-5,5).
  4. 54.Let a and b be real constants such that the function f defined by f(x)= array|c|c| x^2+3x+…
  5. 55.Let f:R R be defined as f(x)=ae^2x+be^x+cx.
  6. 56.The value of n lim _k=1^nn^3(n^2+k^2)(n^2+3k^2) is :
  7. 57.Let f:[-( )/(2),( )/(2)] R be a differentiable function such that f(0)=(1)/(2).
  8. 58.x ( )/(2)lim((|1(x-( )/(2))^2 _x^3^(( )/(2))^3 (t^(1)/(3))dt))| is equal to
  9. 59.If the value of the integral _-( )/(2)^( )/(2)(x^2 x1+ ^x+1+ ^2x1+e^ x^2123)dx=( )/(4)( +a…
  10. 60.For 0<a<1, the value of the integral _0^ dx1-2a x+a^2 is :
  11. 61.If _0^11√3+x+√1+xdx=a+b√2+c√3, where a,b,c are rational numbers, then 2a+3b-4c is equal to…
  12. 62.If (a, b) be the orthocentre of the triangle whose vertices are (1,2),(2,3) and (3,1)
  13. 63.If _0^11(5+2x-2x^2)(1+e^(2-4x))dx=(1)/( ) _e(( +1)/( )), , >0, then ^4- ^4 is equal to :
  14. 64.The value of e^-( )/(4)+ _0^( )/(4)e^-x ^50xdx _0^( )/(4)e^-x( ^49x+ ^51x)dx is
  15. 65.Among (S1): n lim1n^2(2+4+6+ +2n)=1 (S2) : n lim1n^16(1^15+2^15+3^15+ +n^15)=(1)/(16)
  16. 66._0^6e^3x+6e^2x+11e^x+6dx=
  17. 67.If f:R R be a continuous function satisfying _0^( )/(2)f( 2x) xdx+ _0^( )/(4)f( 2x) xdx=0,…
  18. 68.Let the function f:[0,2] R be defined as f(x)= array|c|c| e^minx^2,x-[x], & x [0,1) e^[x- …
  19. 69.The value of the integral _- _e2^ _e2e^x( _e(e^x+√1+e^2x))dx is equal to :
  20. 70.Let f be a continuous function satisfying _0^t^2(f(x)+x^2)dx=(4)/(3)t^3,>0.
  21. 71.Let f(x) be a function satisfying f(x)+f( -x)= ^2, R.
  22. 72.n lim(2^(1)/(2)-2^(1)/(3))(2^(1)/(2)-2^(1)/(5)) ..(2^(1)/(2)-2^(1)/(2n+1)) is equal to :
  23. 73.Let 5f(x)+4f((1)/(x))=(1)/(x)+3,x>0.
  24. 74.Let >0.
  25. 75.If (x)=1x _( )/(4)^x(4√2 t-3 ^'(t))dt,x>0, then ^'(( )/(4)) is equal to :
  26. 76.n lim(3)/(n)4+(2+(1)/(n))^2+(2+(2)/(n))^2+ +(3-(1)/(n))^2 is equal to :
  27. 77.If _ /6^ /4( (x-( )/(3)) (x+( )/(3))+1)dx= _e(√3-1), then 9 ^2 is equal to ____ .
  28. 78.If = _0^2√3 _2(x^2+4), dx+ _2^4√2x-4, dx, then ^2 is equal to ________.
  29. 79.Let f be a differentiable function satisfying f(x)=1-2x+ _0^xe^(x-t)f(t), dt, x R and let …
  30. 80._r=1^20(| ( _0^rx| x|dx)|) Answer: 210
  31. 81.If f(x) satisfies the relation f(x)=e^x+ _1^0(y+xe^x)f(y)dy, then e+f(0) is equal to ____ …
  32. 82.Let a differentiable function f satisfy the equation _36^0f((tx)/(36))dt=4 f(x).
  33. 83.The number of elements in the set S=x:x [0,100] and _0^xt^2 (x-t)dt=x^2 is ____ Answer: 16
  34. 84.Let [.] be the greatest integer function.
  35. 85.Let [ ] denote the greatest integer function and f(x)=n lim1n^3 _k=1^n[k^23^x].
  36. 86.If _0^14 ^-1(1-2x+4x^2)dx=a, ^-1(2)-b, _e(5), where a,b N, then (2a+b) is equal to ____ .
  37. 87.6 _ ^0|( 3x+ 2x+ x)|dx is equal to ____ .
  38. 88.Let [.] denote the greatest integer function.
  39. 89.If 24 _0^( )/(4)[ |4x-( )/(12)|+[2 x]]dx=2 + , where [ ] denotes the greatest integer func…
  40. 90.If t 0lim( _0^1(3x+5)^tdx)^(1)/(t)=( )/(5e)((8)/(5))^(2)/(3), then is equal to ___________…
  41. 91.Let f:(0,) R be a twice differentiable function.
  42. 92.Let n lim(nn^4+1-2n(n^2+1)n^4+1+nn^4+16-8n(n^2+4)n^4+16 + +nn^4+n^4-2n n^2(n^2+n^2)n^4+n^4…
  43. 93.Let [t] denote the largest integer less than or equal to t.
  44. 94.Let r_k= _1^0(1-x^7)^kdx _1^0(1-x^7)^k+1dx,k N.
  45. 95.If f(t)= _0^ 2xdx1- ^2t ^2x,0<t< , then the value of _0^( )/(2) ^2dtf(t) equals __________…
  46. 96.If the shortest distance between the lines [(x+2)/(2)=(y+3)/(3)=(z-5)/(4)] and [(x-3)/(1)=…
  47. 97.If _( )/(4)^0 ^2x1+ x xdx=(1)/(a) _e((a)/(3))+ b√3, where a,b N, then a+b is equal to ____…
  48. 98.Let f:(0,) R and F(x)= _0^xtf(t)dt.
  49. 99.If _- /2^ /28√2 xdx(1+e^ x)(1+ ^4x)= + _e(3+2√2), where , are integers, then ^2+ ^2 equals…
  50. 100.|120 ^3 _0^ x^2 x x ^4x+ ^4xdx| ________.