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Yukti Vault

Calculus

Indefinite Integration: JEE Main & Advanced PYQs

125 previous year questions on Indefinite Integration from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

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Questions 51-100 of 125

  1. $\int\left\{\frac{(\log x-1)}{1+(\log x)^{2}}\right\}^{2}dx$ is equal to
  2. If the integral $\int\frac{\cos8x+1}{\cot2x-\tan2x}dx=A\cos8x+k$ where $k$ is an arbitrary constant, then $A$ is equal to :
  3. If $\int\frac{dx}{x+x^{7}}=p(x)$, then $\int\frac{x^{6}}{x+x^{7}}dx$ is equal to :
  4. $\int\frac{\sin^{8}x-\cos^{8}x}{\left(1-2\sin^{2}x\cos^{2}x\right)}dx$ is equal to
  5. If $f\left(\frac{x-4}{x+2}\right)=2x+1$, $(x\in\mathbb R-\{1,-2\})$, then $\int f(x)\,dx$ is equal to (where $C$ is a constant of integration)
  6. The integral $\int\frac{2x^{12}+5x^{9}}{\left(x^{5}+x^{3}+1\right)^{3}}dx$ is equal to
  7. The integral $\int\frac{\left(x^8-x^2\right)dx}{\left(x^{12}+3x^6+1\right)\tan^{-1}\left(x^3+\frac1{x^3}\right)}$ is equal to :
  8. Let $f(x)=\int\frac{2x}{(x^{2}+1)(x^{2}+3)}dx$. If $f(3)=\frac12\left(\log_e5-\log_e6\right)$, then $f(4)$ is equal to
  9. If $\int\frac{\sin^{3/2}x+\cos^{3/2}x}{\sqrt{\sin^{3}x\cos^{3}x\sin(x-\theta)}}dx=A\sqrt{\cos\theta\tan x-\sin\theta}+B\sqrt{\cos\theta-\sin\theta\cot x}+C$, where $C$ …
  10. For $x^2\ne n\pi+1$, $n\in N$ (the set of natural numbers), the integral …
  11. If $\int\frac{\sqrt{1-x^{2}}}{x^{4}}dx=A(x)\left(\sqrt{1-x^{2}}\right)^{m}+C$, for a suitable chosen integer $m$ and a function $A(x)$, where $C$ is a constant of …
  12. The integral $\int\frac{dx}{(x+1)^{3/4}(x-2)^{5/4}}$ is equal to
  13. The integral $\int\frac{\sin^2x\cos^2x}{\left(\sin^5x+\cos^3x\sin^2x+\sin^3x\cos^2x+\cos^5x\right)^2}dx$ is equal to (where $C$ is the constant of integration).
  14. If $\int\frac{\log\left(t+\sqrt{1+t^{2}}\right)}{\sqrt{1+t^{2}}}dt=\frac12\left(g(t)\right)^{2}+c$, where $c$ is a constant, then $g(2)$ is equal to
  15. $\int\frac{\sin\frac{5x}{2}}{\sin\frac x2}dx$ is equal to (where $c$ is a constant of integration)
  16. If $\int\frac{x+1}{\sqrt{2x-1}}dx=f(x)\sqrt{2x-1}+C$, where $C$ is a constant of integration, then $f(x)$ is equal to :
  17. If $f\left(\frac{3x-4}{3x+4}\right)=x+2$, $x\neq-\frac43$, and $\int f(x)dx=A\log|1-x|+Bx+C$, then the ordered pair $(A,B)$ is equal to
  18. Let $n\ge2$ be a natural number and $0<\theta<\frac\pi2$. Then $\int\frac{\left(\sin^{n}\theta-\sin\theta\right)^{\frac1n}\cos\theta}{\sin^{n+1}\theta}d\theta$ is equal …
  19. If $\int x^5e^{-4x^3}dx=\frac1{48}e^{-4x^3}f(x)+C$, where $C$ is a constant of integration, then $f(x)$ is equal to :
  20. If $\int\frac{dx}{\left(x^{2}-2x+10\right)^{2}}=A\left(\tan^{-1}\left(\frac{x-1}{3}\right)+\frac{f(x)}{x^{2}-2x+10}\right)+C$, then (where $C$ is a constant of …
  21. Let $I_n=\int\tan^nx\,dx$, $(n>1)$. If $I_4+I_6=a\tan^5x+bx^5+c$, then the ordered pair $(a,b)$ is equal to :
  22. If $\int\frac{dx}{x^3\left(1+x^6\right)^{2/3}}=xf(x)\left(1+x^6\right)^{1/3}+C$, where $C$ is a constant of integration, then the function $f(x)$ is equal to :
  23. The integral $\int\frac{dx}{(1+\sqrt x)\sqrt{x-x^{2}}}$ is equal to
  24. $\int\sec^{2}x\cdot\cot^{4/3}x\,dx$ is equal to
  25. The integral $\int\frac{2x^3-1}{x^4+x}dx$ is equal to
  26. If $m$ is a non-zero number and $\int\frac{x^{5m-1}+2x^{4m-1}}{\left(x^{2m}+x^{m}+1\right)^{3}}dx=f(x)+c$, then $f(x)$ is equal to
  27. If $f(x)=\int\frac{5x^{8}+7x^{6}}{\left(x^{2}+1+2x^{7}\right)^{2}}dx$, $(x\ge0)$, and $f(0)=0$, then the value of $f(1)$ is
  28. The integral $\int\sqrt{1+2\cot x(\operatorname{cosec}x+\cot x)}\,dx$, $\left(0<x<\frac\pi2\right)$ is equal to :
  29. The integral $\int\frac{\sin^{2}x\cos^{2}x}{\left(\sin^{3}x+\cos^{3}x\right)^{2}}dx$ is equal to :
  30. If $\int e^{\sec x}\left(\sec x\tan x\,f(x)+\left(\sec x\tan x+\sec^{2}x\right)\right)dx=e^{\sec x}f(x)+C$, then a possible choice of $f(x)$ is :
  31. The integral $\int x\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)dx\ (x>0)$ is equal to :
  32. $\int\frac{2x+5}{\sqrt{7-6x-x^{2}}}dx=A\sqrt{7-6x-x^{2}}+B\sin^{-1}\left(\frac{x+3}{4}\right)+C$ (where $C$ is a constant of integration), then the ordered pair $(A,B)$ …
  33. The integral $\int\frac{3x^{13}+2x^{11}}{(2x^{4}+3x^{2}+1)^{4}}dx$ is equal to
  34. If $\int\frac{dx}{\cos^{3}x\sqrt{2\sin2x}}=(\tan x)^{A}+C(\tan x)^{B}+k$, where $k$ is a constant of integration, then $A+B+C$ equals
  35. The integral $\int\cos(\ln x)\,dx$ is equal to
  36. If $\int x^{5}e^{-x^{2}}dx=g(x)e^{-x^{2}}+c$, where $c$ is a constant of integration, then $g(-1)$ is equal to
  37. If $\int\frac{\tan x}{1+\tan x+\tan^{2}x}dx=x-\frac{K}{\sqrt A}\tan^{-1}\left(\frac{K\tan x+1}{\sqrt A}\right)+C$ ($C$ is a constant of integration), then the ordered …
  38. Let $\alpha\in\left(0,\frac\pi2\right)$ be constant. If the integral $\int\frac{\tan x+\tan\alpha}{\tan x-\tan\alpha}dx=A(x)\cos2\alpha+B(x)\sin2\alpha+C$, where $C$ is …
  39. Evaluate: $\displaystyle\int\frac{(x-1)e^{x}}{(x+1)^{3}}\,dx$.
  40. Evaluate $\displaystyle\int\frac{\sin x}{\sin x-\cos x}\,dx$.
  41. Evaluate $\displaystyle\int\left(e^{\log x}+\sin x\right)\cos x\,dx$.
  42. If $\int\sqrt{\sec2x-1}\,dx=\alpha\log_e\left|\cos2x+\beta+\sqrt{\cos2x\left(1+\cos\frac1\beta x\right)}\right|+\text{constant}$, then $\beta-\alpha$ is equal to
  43. Evaluate $\displaystyle\int\frac{x^{2}\,dx}{(a+bx)^{2}}$.
  44. Evaluate: $\displaystyle\int\sqrt{\frac{1-\sqrt x}{1+\sqrt x}}\,dx$.
  45. Evaluate the following: $\displaystyle\int\frac{dx}{x^{2}\left(x^{4}+1\right)^{3/4}}$
  46. Evaluate: $\displaystyle\int\frac{(\cos2x)^{1/2}}{\sin x}\,dx$.
  47. Find the indefinite integral $\displaystyle\int\cos2\theta\,\ln\left(\dfrac{\cos\theta+\sin\theta}{\cos\theta-\sin\theta}\right)d\theta$.
  48. If $\displaystyle\int\frac{4e^{x}+6e^{-x}}{9e^{x}-4e^{-x}}\,dx=Ax+B\log\left(9e^{2x}-4\right)+C$, then $A=$ ......, $B=$ ..... and $C=$ .....
  49. Evaluate $\displaystyle\int\left(\sqrt{\tan x}+\sqrt{\cot x}\right)dx$.
  50. The value of the integral $\displaystyle\int\frac{\cos^{3}x+\cos^{5}x}{\sin^{2}x+\sin^{4}x}\,dx$ is

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