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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 1-50 of 125

  1. The integral $\int\frac{e^{3\log_e2x}+5e^{2\log_e2x}}{e^{4\log_ex}+5e^{3\log_ex}-7e^{2\log_ex}}dx$, $x>0$, is equal to : (where $c$ is a constant of integration)
  2. The integral $\int\frac{dx}{\sqrt{2x+5}-\sqrt{2x+3}}$ is equal to (where $C$ is a constant of integration)
  3. The value of the integral …
  4. The integral $\int\frac{(2x-1)\cos\sqrt{(2x-1)^{2}+5}}{\sqrt{4x^{2}-4x+6}}dx$ is equal to : (where $c$ is a constant of integration)
  5. Let $f(x)=\int\left(\frac{16x+24}{x^{2}+2x-15}\right)dx$. If $f(4)=14\log_e(3)$ and $f(7)=\log_e(2^{\alpha}\cdot 3^{\beta})$, $\alpha,\beta\in\mathbf{N}$, then …
  6. If $\int\frac{\cos x-\sin x}{\sqrt{8-\sin2x}}dx=a\sin^{-1}\left(\frac{\sin x+\cos x}{b}\right)+c$, where $c$ is a constant of integration, then the ordered pair $(a,b)$ …
  7. If $\int\frac{4x+13}{\sqrt{x^{2}+5x+6}}dx=A\sqrt{x^{2}+5x+6}+B\log_e\left|\left(x+\frac{5}{2}\right)+\sqrt{x^{2}+5x+6}\right|+C$ ($C$ is a constant of integration), then …
  8. Let $I(x)=\int\frac{(x+1)}{x\left(1+xe^{x}\right)^{2}}dx$, $x>0$. If $\lim_{x\to\infty}I(x)=0$, then $I(1)$ is equal to
  9. For $x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$, if $y(x)=\int\frac{\operatorname{cosec}x+\sin x}{\operatorname{cosec}x\sec x+\tan x\sin^{2}x}dx$, and …
  10. For $\alpha,\beta,\gamma,\delta\in\mathbb{N}$, if …
  11. Let $I(x)=\int\frac{x^{2}\left(x\sec^{2}x+\tan x\right)}{(x\tan x+1)^{2}}dx$. If $I(0)=0$, then $I\left(\frac{\pi}{4}\right)$ is equal to
  12. If $\int\frac{1}{a^{2}\sin^{2}x+b^{2}\cos^{2}x}dx=\frac{1}{12}\tan^{-1}(3\tan x)+\text{constant}$, then the maximum value of $a\sin x+b\cos x$, is :
  13. The integral $\int\frac{1}{\sqrt[4]{(x-1)^{3}(x+2)^{5}}}dx$ is equal to : (where $C$ is a constant of integration)
  14. Let $I(x)=\int\frac{6}{\sin^{2}x(1-\cot x)^{2}}dx$. If $I(0)=3$, then $I\left(\frac{\pi}{12}\right)$ is equal to
  15. Let $\int\frac{2-\tan x}{3+\tan x}dx=\frac{1}{2}\left(\alpha x+\log_e|\beta\sin x+\gamma\cos x|\right)+C$, where $C$ is the constant of integration. Then …
  16. If $I(x)=\int e^{\sin^{2}x}\left(\cos x\sin2x-\sin x\right)dx$ and $I(0)=1$, then $I\left(\frac{\pi}{3}\right)$ is equal to
  17. Let $f(x)=\int x^{3}\sqrt{3-x^{2}}\,dx$. If $5f(\sqrt{2})=-4$, then $f(1)$ is equal to
  18. If $f'(x)=\tan^{-1}(\sec x+\tan x)$, $-\frac\pi2<x<\frac\pi2$, and $f(0)=0$, then $f(1)$ is equal to :
  19. For real numbers $\alpha,\beta,\gamma$ and $\delta$, if …
  20. If $\int\frac{d\theta}{\cos^2\theta(\tan2\theta+\sec2\theta)}=\lambda\tan\theta+2\log_e|f(\theta)|+C$ where $C$ is a constant of integration, then the ordered pair …
  21. If $\int\frac{\cos x\,dx}{\sin^3x\left(1+\sin^6x\right)^{2/3}}=f(x)\left(1+\sin^6x\right)^{1/\lambda}+c$ where $c$ is a constant of integration, then …
  22. The integral $\int\frac{\left(1-\frac{1}{\sqrt3}\right)(\cos x-\sin x)}{\left(1+\frac{2}{\sqrt3}\sin2x\right)}dx$ is equal to
  23. If $\int\frac{\left(x^{2}+1\right)e^{x}}{(x+1)^{2}}dx=f(x)e^{x}+C$, where $C$ is a constant, then $\frac{d^{3}f}{dx^{3}}$ at $x=1$ is equal to :
  24. If $\int(e^{2x}+2e^x-e^{-x}-1)e^{(e^x+e^{-x})}dx=g(x)e^{(e^x+e^{-x})}+c$, where $c$ is a constant of integration, then $g(0)$ is equal to :
  25. The integral $\int\left(\frac{x}{x\sin x+\cos x}\right)^2dx$ is equal to (where $C$ is a constant of integration) :
  26. If $\int\frac{1}{x}\sqrt{\frac{1-x}{1+x}}dx=g(x)+c$, $g(1)=0$, then $g\left(\frac{1}{2}\right)$ is equal to :
  27. Let $f(x)=\int\frac{\sqrt x}{(1+x)^2}dx$ ($x\ge0$). Then $f(3)-f(1)$ is equal to :
  28. The integral $\int\frac{dx}{(x+4)^{8/7}(x-3)^{6/7}}$ is equal to : (where $C$ is a constant of integration)
  29. If $\int\sin^{-1}\left(\sqrt{\frac x{1+x}}\right)dx=A(x)\tan^{-1}(\sqrt x)+B(x)+C$, where $C$ is a constant of integration, then the ordered pair $(A(x),B(x))$ can be :
  30. For $I(x)=\int\frac{\sec^2x-2022}{\sin^{2022}x}dx$, if $I\left(\frac\pi4\right)=2^{1011}$, then
  31. Let $I(x)=\int\sqrt{\frac{x+7}{x}}\,dx$ and $I(9)=12+7\log_e7$. If $I(1)=\alpha+7\log_e\left(1+2\sqrt{2}\right)$, then $\alpha^{4}$ is equal to $\_\_\_\_\_\_$ .
  32. If …
  33. If …
  34. If $\int\frac{1}{\sqrt[5]{(x-1)^{4}(x+3)^{6}}}dx=A\left(\frac{\alpha x-1}{\beta x+3}\right)^{B}+C$, where $C$ is the constant of integration, then the value of …
  35. If $\int\frac{2e^{x}+3e^{-x}}{4e^{x}+7e^{-x}}dx=\frac{1}{14}\left(ux+v\log_e\left(4e^{x}+7e^{-x}\right)\right)+C$, where $C$ is a constant of integration, then $u+v$ is …
  36. If $\int\frac{dx}{\left(x^{2}+x+1\right)^{2}}=a\tan^{-1}\left(\frac{2x+1}{\sqrt{3}}\right)+b\left(\frac{2x+1}{x^{2}+x+1}\right)+C$, $x>0$ where $C$ is the constant of …
  37. If $f(x)=\int\frac{5x^{8}+7x^{6}}{\left(x^{2}+1+2x^{7}\right)^{2}}dx$, $(x\ge0)$, $f(0)=0$ and $f(1)=\frac{1}{K}$, then the value of $K$ is $\_\_\_\_\_\_$ .
  38. If $\int\left(\sqrt{\tan x}+\sqrt{\cot x}\right)dx=a\tan^{-1}\left(\frac{\tan x-1}{\sqrt{b\tan x}}\right)+C$, then the value of $a^{4}+b^{5}$ is $\_\_\_\_\_\_$ . (Here …
  39. If $\int\frac{\sin x}{\sin^{3}x+\cos^{3}x}dx=\alpha\log_e|1+\tan x|+\beta\log_e\left|1-\tan x+\tan^{2}x\right|+\gamma\tan^{-1}\left(\frac{2\tan x-1}{\sqrt{3}}\right)+C$ …
  40. Let $f(x)=\int\frac{dx}{\left(3+4x^{2}\right)\sqrt{4-3x^{2}}}$, $|x| 0$, then $\alpha^{2}+\beta^{2}$ is equal to $\_\_\_\_\_\_$ .
  41. $\int\frac{dx}{\cos x+\sqrt3\sin x}$ equals
  42. If $\int\frac{\sin x}{\sin(x-\alpha)}\,dx=Ax+B\log\sin(x-\alpha)+C$, then value of $(A,B)$ is
  43. If $\int\frac{x^{2}-x+1}{x^{2}+1}e^{\cot^{-1}x}dx=A(x)e^{\cot^{-1}x}+C$, then $A(x)$ is equal to :
  44. If $\int f(x)\,dx=\psi(x)$, then $\int x^{5}f\left(x^{3}\right)dx$ is equal to
  45. The value of $\sqrt2\int\frac{\sin x\,dx}{\sin\left(x-\frac\pi4\right)}$ is
  46. The integral $\int\frac{x\,dx}{2-x^{2}+\sqrt{2-x^{2}}}$ equals :
  47. $\int\frac{dx}{\cos x-\sin x}$ is equal to
  48. The integral $\int\left(1+x-\frac1x\right)e^{x+\frac1x}dx$ is equal to
  49. At present, a firm is manufacturing $2000$ items. It is estimated that the rate of change of production $P$ w.r.t. additional number of workers $x$ is given by …
  50. The integral $\int\frac{dx}{x^{2}\left(x^{4}+1\right)^{3/4}}$ equals :

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