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Indefinite Integrals

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  1. 51.Let [] denote the greatest integer function.
  2. 52.Let f(x) = 7^8x+7^6x-3^4x-3^2x, and let I_1=_0^/4f(x),dx, I_2=_0^/4xf(x),dx.
  3. 53.If I=_e^2^e^41xe^1/(( x)^2+1)e^1/(( x)^2+1)+e^1/((6- x)^2+1),dx, then the value of I is:
  4. 54.If P=_0^/2^3/2x^3/2x+^3/2x,dx (so P=/4 by King's rule), then _0^2P(x x x)/(^4x+^4x),dx equ…
  5. 55.Let I(m,n)=_0^1x^m-1(1-x)^n-1,dx, m,n>0.
  6. 56._-/2^/2(96x^2^2x)/(1+e^x),dx = (^2+), ,Z.
  7. 57.Let f and g be functions with g(x)=_0^x t,f(t),dt satisfying g(x^3) = x^6+x^7.
  8. 58.Let f be a twice differentiable function such that f(2)=1.
  9. 59.4_0^1dx√3+x^2+√1+x^2 - 3_e(√3) is equal to:
  10. 60.Let (a,b) be the point of intersection of the curve x^2=2y and the straight line y-2x-6=0 …
  11. 61.Let f:[1,)[2,) be a differentiable function satisfying 10_1^x f(t),dt = 5xf(x)-x^5-9 for x…
  12. 62.Let the domain of f(x)=_2_4_6(3+4x-x^2) be (a,b).
  13. 63.The integral _0^ (8x)/(4^2x+^2x),dx is equal to:
  14. 64.Let f(x)+2f(1x)=x^2+5 and 2g(x)-3g(1x)=x, x>0.
  15. 65.The integral _0^((x+3) x)/(1+3^2x),dx is equal to:
  16. 66.e^x(x^-1x√1-x^2+^-1x(1-x^2)^3/2+(x)/(1-x^2))dx = g(x)+C.
  17. 67.Let I(x) = dx(x-11)^11/13(x+15)^15/13.
  18. 68.x^3 x,dx = g(x)+C.
  19. 69.f(x) = 1x^1/4(1+x^1/4),dx, f(0)=-6.
  20. 70.Let f(x) = x^3√3-x^2,dx.
  21. 71.2x^2+5x+9x^2+x+1,dx = xx^2+x+1+x^2+x+1+_e|x+12+x^2+x+1|+C, where C is the constant of inte…
  22. 72.(√1+x^2+x)^10(√1+x^2-x)^9,dx = 1m[(√1+x^2+x)^n(n√1+x^2-x)]+C, where C is the constant of i…
  23. 73.(1x^2+1x^4)([3]x+1x^3)^1/23dx = -3(+1)(3x^+x^)^(+1)/()+C, x>0, ,,Z, where C is the constan…
  24. 74.The value of the integral simplifies to:
  25. 75.The value of 8_0^/2( x)^2023( x)^2023+( x)^2023,dx is ____ Answer: 2
  26. 76.3√34_3√2/4^48√9-4x^2,dx --- i.e.
  27. 77.Let f be a differentiable function defined on (0,2) such that f(x)>0 and [f(x) + _0^x f(t)…
  28. 78.The minimum value of the function f(x) = _0^2 e^|x-t|,dt is:
  29. 79.The value of _n1+2-3+4+5-6++(3n-2)+(3n-1)-3n√2n^4+4n+3-n^4+5n+4 is:
  30. 80.The integral 16_1^2(dx)/(x^3(x^2+2)^2) is equal to:
  31. 81.If _1/3^3 |_ex|,dx = mn_e(n^2e), where m and n are coprime natural numbers, then m^2+n^2-5…
  32. 82.Let [x] denote the greatest integer x.
  33. 83.Let f(x) = x+a^2-4 x+b^2-4 x, xR, be a function satisfying f(x) = x+_0^/2(x+y)f(y),dy.
  34. 84.The value of the integral _1/2^2^-1xx,dx is equal to:
  35. 85.The value of the integral _1^2(t^4+1)/(t^6+1),dt is:
  36. 86.If [x] denotes the greatest integer x, then the value of _1^2 x^2,e^[x]+[x^3],dx is:
  37. 87._x0(48)/(x^4)_0^x(t^3)/(t^6+1),dt is equal to _______.
  38. 88._n3n_r=0^n-1(2+(r)/(n))^2 is equal to:
  39. 89.The value of _/3^/2(2+3 x)/( x(1+ x)),dx is equal to:
  40. 90.Let a differentiable function f satisfy f(x)+_3^x(f(t))/(t),dt = x+1, x3.
  41. 91.Let (0,1) and =_e(1-).
  42. 92.Let f(x) = (2x)/((x^2+1)(x^2+3)),dx.
  43. 93.If 2x-1,dx = _e|2x+2x√1+2x|+ x + constant, then - is equal to _______.
  44. 94._n[11+n+12+n+13+n++12n] is equal to:
  45. 95.If _0^1(x^21+x^14+x^7)(2x^14+3x^7+6)^1/7,dx = 1l(11)^m/n, where l,m,nN, m,n coprime, then …
  46. 96.The value of the integral _-/4^/4(x+/4)/(2-2x),dx is:
  47. 97.If _0^5^ x1+5^ x(1+ x3x+^2x+^3x3x)dx = k16, then k is equal to ______.
  48. 98.Let f(x) be a function satisfying f(x)+f(-x)=^2, xR.
  49. 99.Let f be a continuous function satisfying _0^t^2(f(x)+x^2),dx = 43t^3, t>0.
  50. 100.The value of the integral _-_e2^_e2e^x_e(e^x+√1+e^2x)dx is equal to: