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Yukti Vault

Calculus

Definite Integration: JEE Main & Advanced PYQs

392 previous year questions on Definite 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 392

  1. $\int_{1}^{2}\left([x^{2}-1]+[x^{2}+1]\right)dx$ is equal to ($[t]$ denotes the greatest integer $\le t$)
  2. The value of the integral $\int_{1}^{3}\left[x^{2}-2x-2\right]dx$, where $[x]$ denotes the greatest integer less than or equal to $x$, is :
  3. The value of $\int_{-1}^{1}x^{2}e^{[x^{3}]}dx$, where $[t]$ denotes the greatest integer $\le t$, is :
  4. If $\int_{0}^{100\pi}\frac{\sin^{2}x}{e^{\left(\frac{x}{\pi}-\left[\frac{x}{\pi}\right]\right)}}dx=\frac{\alpha\pi^{3}}{1+4\pi^{2}}$, $\alpha\in\mathbf{R}$, where $[x]$ …
  5. Let $g(x)=\int_{0}^{x}f(t)\,dt$, where $f$ is continuous function in $[0,3]$ such that $\frac{1}{3}\le f(t)\le1$ for all $t\in[0,1]$ and $0\le f(t)\le\frac{1}{2}$ for …
  6. Let $f:\mathbf{R}\to\mathbf{R}$ be defined as $f(x)=e^{-x}\sin x$. If $F:[0,1]\to\mathbf{R}$ is a differentiable function such that $F(x)=\int_{0}^{x}f(t)\,dt$, then the …
  7. The value of $\int_{0}^{20\pi}(\sin^{4}x+\cos^{4}x)\,dx$ is equal to :
  8. The integral $\int_{0}^{1}\cot^{-1}\left(1+x+x^{2}\right)dx$ is equal to:
  9. The value of $\int_{-\pi/2}^{\pi/2}\frac{\cos^{2}x}{1+3^{x}}dx$ is :
  10. The value of $\sum_{n=1}^{100}\int_{n-1}^{n}e^{x-[x]}dx$, where $[x]$ is the greatest integer $\le x$, is :
  11. The value of the integral $\int_{-1}^{1}\left(\frac{x^{3}+|x|+1}{x^{2}+2|x|+1}\right)dx$ is equal to :
  12. For $x>0$, if $f(x)=\int_{1}^{x}\frac{\log_et}{(1+t)}dt$, then $f(e)+f\left(\frac{1}{e}\right)$ is equal to :
  13. Let $f(x)$ be a differentiable function defined on $[0,2]$ such that $f'(x)=f'(2-x)$ for all $x\in(0,2)$, $f(0)=1$ and $f(2)=e^{2}$. Then the value of …
  14. The value of the definite integral $\int_{\pi/24}^{5\pi/24}\frac{dx}{1+\sqrt[3]{\tan2x}}$ is :
  15. The value of $\int_{-1/\sqrt{2}}^{1/\sqrt{2}}\left(\left(\frac{x+1}{x-1}\right)^{2}+\left(\frac{x-1}{x+1}\right)^{2}-2\right)^{1/2}dx$ is :
  16. Let $\left(2^{1-a}+2^{1+a}\right)$, $f(a)$, $\left(3^{a}+3^{-a}\right)$ be in A.P. and $\alpha$ be the minimum value of $f(a)$. Then the value of the integral …
  17. Consider the integral $I=\int_{0}^{10}\frac{[x]e^{[x]}}{e^{x-1}}dx$, where $[x]$ denotes the greatest integer less than or equal to $x$. Then the value of $I$ is equal …
  18. The value of $\int_{-\pi/2}^{\pi/2}\left(\frac{1+\sin^{2}x}{1+\pi^{\sin x}}\right)dx$ is :
  19. If $I_n=\int_{\pi/4}^{\pi/2}\cot^{n}x\,dx$, then :
  20. If $I_1=\int_{0}^{\pi}xe^{\operatorname{cosec}x}dx$ and $I_2=\int_{0}^{\pi}e^{\operatorname{cosec}x}dx$, then the ratio $I_1:I_2$ is equal to
  21. The value of the integral $\int_{0}^{2}\frac{\sqrt{x\left(x^{2}+x+1\right)}}{\left(\sqrt{x+1}\right)\left(\sqrt{x^{4}+x^{2}+1}\right)}\,dx$ is equal to :
  22. If the integral $\int_{0}^{10}\frac{[\sin2\pi x]}{e^{x-[x]}}dx=\alpha e^{-1}+\beta e^{-\frac{1}{2}}+\gamma$, where $\alpha,\beta,\gamma$ are integers and $[x]$ denotes …
  23. The value of the integral $\int_{-1}^{1}\log\left(x+\sqrt{x^{2}+1}\right)dx$ is :
  24. Let $f(x)$ be a positive function and $I_1=\int_{-\frac{1}{2}}^{1}2xf(2x(1-2x))\,dx$ and $I_2=\int_{-1}^{2}f(x(1-x))\,dx$. Then the value of $\frac{I_2}{I_1}$ is equal to
  25. If $(a,b)$ be the orthocentre of the triangle whose vertices are $(1,2)$, $(2,3)$ and $(3,1)$, and $I_1=\int_{a}^{b}x\sin\left(4x-x^{2}\right)dx$ …
  26. If $[x]$ is the greatest integer $\le x$, then $\pi^{2}\int_{0}^{2}\left(\sin\frac{\pi x}{2}\right)(x-[x])^{[x]}dx$ is equal to :
  27. The value of the integral $\int_{-\pi/2}^{\pi/2}\frac{dx}{(1+e^{x})(\sin^{6}x+\cos^{6}x)}$ is equal to
  28. The value of $\int_{0}^{\pi}\frac{e^{\cos x}\sin x}{\left(1+\cos^{2}x\right)\left(e^{\cos x}+e^{-\cos x}\right)}dx$ is equal to :
  29. The value of the integral $\int_{-\log_e2}^{\log_e2}e^{x}\left(\log_e\left(e^{x}+\sqrt{1+e^{2x}}\right)\right)dx$ is equal to
  30. $4\int_{0}^{1}\left(\frac{1}{\sqrt{3+x^{2}}+\sqrt{1+x^{2}}}\right)dx-3\log_e(\sqrt{3})$ is equal to :
  31. The value of the integral $\int_{0}^{1}\frac{\sqrt{x}\,dx}{(1+x)(1+3x)(3+x)}$ is :
  32. The integral $\int_{1/4}^{3/4}\cos\left(2\cot^{-1}\sqrt{\frac{1-x}{1+x}}\right)dx$ is equal to
  33. The value of the intergral $\int_{0}^{\pi/4}\frac{x\,dx}{\sin^{4}(2x)+\cos^{4}(2x)}$ equals :
  34. Let $f(x)+2f\left(\frac{1}{x}\right)=x^{2}+5$ and $2g(x)-3g\left(\frac{1}{2}\right)=x$, $x>0$. If $\alpha=\int_{1}^{2}f(x)\,dx$, and $\beta=\int_{1}^{2}g(x)\,dx$, then …
  35. If $f:\mathbb{R}\to\mathbb{R}$ be a continuous function satisfying $\int_{0}^{\pi/2}f(\sin2x)\sin x\,dx+\alpha\int_{0}^{\pi/4}f(\cos2x)\cos x\,dx=0$, then the value of …
  36. If the value of the integral $\int_{0}^{5}\frac{x+[x]}{e^{x-[x]}}dx=\alpha e^{-1}+\beta$, where $\alpha,\beta\in\mathbf{R}$, $5\alpha+6\beta=0$, and $[x]$ denotes the …
  37. If $\int_{0}^{1}\frac{1}{\left(5+2x-2x^{2}\right)\left(1+e^{(2-4x)}\right)}dx=\frac{1}{\alpha}\log_e\left(\frac{\alpha+1}{\beta}\right)$, $\alpha,\beta>0$, then …
  38. Let $f,g:(0,\infty)\to\mathbb{R}$ be two functions defined by $f(x)=\int_{-x}^{x}\left(|t|-t^{2}\right)e^{-t^{2}}dt$ and $g(x)=\int_{0}^{x^{2}}t^{1/2}e^{-t}dt$. Then …
  39. The value of $\frac{e^{-\frac{\pi}{4}}+\int_{0}^{\pi/4}e^{-x}\tan^{50}x\,dx}{\int_{0}^{\pi/4}e^{-x}\left(\tan^{49}x+\tan^{51}x\right)dx}$ is
  40. $\int_{6}^{16}\frac{\log_ex^{2}}{\log_ex^{2}+\log_e\left(x^{2}-44x+484\right)}dx$ is equal to :
  41. The value of the integral $\int_{-1}^{2}\log_e\left(x+\sqrt{x^{2}+1}\right)dx$ is
  42. Let $5f(x)+4f\left(\frac{1}{x}\right)=\frac{1}{x}+3$, $x>0$. Then $18\int_{1}^{2}f(x)\,dx$ is equal to:
  43. Let the function $f:[0,2]\to\mathbb{R}$ be defined as $f(x)=\begin{cases}e^{\min\{x^{2},x-[x]\}}, & x\in[0,1)\\ e^{[x-\log_ex]}, & x\in[1,2]\end{cases}$ where $[t]$ …
  44. Let $y=f(x)$ be a thrice differentiable function in $(-5,5)$. Let the tangents to the curve $y=f(x)$ at $(1,f(1))$ and $(3,f(3))$ make angles $\pi/6$ and $\pi/4$ …
  45. Let $(a,b)$ be the point of intersection of the curve $x^{2}=2y$ and the straight line $y-2x-6=0$ in the second quadrant. Then the integral …
  46. Let $\int_{\alpha}^{\log_e4}\frac{dx}{\sqrt{e^{x}-1}}=\frac{\pi}{6}$. Then $e^{\alpha}$ and $e^{-\alpha}$ are the roots of the equation :
  47. Let $f(x)$ be a function satisfying $f(x)+f(\pi-x)=\pi^{2}$, $\forall x\in\mathbb{R}$. Then $\int_{0}^{\pi}f(x)\sin x\,dx$ is equal to
  48. Let $f:\mathbb{R}\to\mathbb{R}$ be defined as $f(x)=ae^{2x}+be^{x}+cx$. If $f(0)=-1$, $f'(\log_e2)=21$ and $\int_{0}^{\log_e4}(f(x)-cx)\,dx=\frac{39}{2}$, then the value …
  49. The value of $k\in\mathbb{N}$ for which the integral $I_n=\int_{0}^{1}\left(1-x^{k}\right)^{n}dx$, $n\in\mathbb{N}$, satisfies $147I_{20}=148I_{21}$ is
  50. $\int_{0}^{\infty}\frac{6}{e^{3x}+6e^{2x}+11e^{x}+6}dx=$

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