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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 51-100 of 392

  1. Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined by $f(x)=\frac{x}{\left(1+x^{4}\right)^{1/4}}$, and $g(x)=f(f(f(f(x))))$. Then …
  2. If $\int_{0}^{\pi/3}\cos^{4}x\,dx=a\pi+b\sqrt{3}$, where $a$ and $b$ are rational numbers, then $9a+8b$ is equal to :
  3. The value of $\int_{-1}^{1}\frac{\left(1+\sqrt{|x|-x}\right)e^{x}+\left(\sqrt{|x|-x}\right)e^{-x}}{e^{x}+e^{-x}}dx$ is equal to
  4. $\int_{0}^{\pi/4}\frac{\cos^{2}x\sin^{2}x}{\left(\cos^{3}x+\sin^{3}x\right)^{2}}dx$ is equal to
  5. Let $a$ and $b$ be real constants such that the function $f$ defined by $f(x)=\begin{cases}x^{2}+3x+a, & x\le1\\ bx+2, & x>1\end{cases}$ be differentiable on …
  6. If $\int_{0}^{1}\frac{1}{\sqrt{3+x}+\sqrt{1+x}}dx=a+b\sqrt{2}+c\sqrt{3}$, where $a,b,c$ are rational numbers, then $2a+3b-4c$ is equal to :
  7. The integral $\int_{-1}^{\frac{3}{2}}\left(\left|\pi^{2}x\sin(\pi x)\right|\right)dx$ is equal to :
  8. The integral $\int_{0}^{\pi}\frac{8x\,dx}{4\cos^{2}x+\sin^{2}x}$ is equal to
  9. Let $f$ be a continuous function satisfying $\int_{0}^{t^{2}}\left(f(x)+x^{2}\right)dx=\frac{4}{3}t^{3}$, $\forall t>0$. Then $f\left(\frac{\pi^{2}}{4}\right)$ is equal …
  10. The value of $\int_{0}^{1}\left(2x^{3}-3x^{2}-x+1\right)^{\frac{1}{3}}dx$ is equal to :
  11. If the value of the integral $\int_{-\pi/2}^{\pi/2}\left(\frac{x^{2}\cos x}{1+\pi^{x}}+\frac{1+\sin^{2}x}{1+e^{\sin x^{2023}}}\right)dx=\frac{\pi}{4}(\pi+a)-2$, then the …
  12. The integral $\int_{0}^{\pi}\frac{(x+3)\sin x}{1+3\cos^{2}x}dx$ is equal to
  13. If for all real triplets $(a,b,c)$, $f(x)=a+bx+cx^2$; then $\int_0^1f(x)\,dx$ is equal to :
  14. Let $A_k=\int_{k\pi}^{(k+1)\pi}e^{-x}\sin x\,dx$, $k=0,1,2,\ldots,20$. Then $\sum_{k=0}^{20}|A_k|$ is equal to
  15. Let $f:\mathbf{R}\to\mathbf{R}$ be a continuous function satisfying $f(x)+f(x+k)=n$, for all $x\in\mathbf{R}$ where $k>0$ and $n$ is a positive integer. If …
  16. Let $f(x)$ and $g(x)$ be two functions satisfying $f\left(x^{2}\right)+g(4-x)=4x^{3}$ and $g(4-x)+g(x)=0$, then the value of $\int_{-4}^{4}f\left(x^{2}\right)dx$ is …
  17. The value of $\int_{-\pi/2}^{\pi/2}\frac1{1+e^{\sin x}}dx$ is :
  18. The value of the integral $\int_{-2}^{2}\frac{\left|x^{3}+x\right|}{\left(e^{x|x|}+1\right)}dx$ is equal to :
  19. The integral $\int_{\pi/6}^{\pi/3}\tan^3x\cdot\sin^23x\,(2\sec^2x\cdot\sin^23x+3\tan x\cdot\sin6x)\,dx$ is equal to :
  20. Let $f(x)=|x-2|$ and $g(x)=f(f(x))$, $x\in[0,4]$. Then $\int_0^3(g(x)-f(x))\,dx$ is equal to :
  21. The value of $\lim_{n\to\infty}\frac1n\sum_{j=1}^{n}\frac{(2j-1)+8n}{(2j-1)+4n}$ is equal to :
  22. The integral $\int_{0}^{1}\frac{1}{7^{\left[\frac{1}{x}\right]}}dx$, where $[\cdot]$ denotes the greatest integer function, is equal to
  23. If $\int_{\pi/6}^{\pi/4}\left(\cot\left(x-\frac{\pi}{3}\right)\cot\left(x+\frac{\pi}{3}\right)+1\right)dx=\alpha\log_e\left(\sqrt{3}-1\right)$, then $9\alpha^{2}$ is …
  24. If $I=\int_1^2\frac{dx}{\sqrt{2x^3-9x^2+12x+4}}$, then :
  25. For any real number $x$, let $[x]$ denote the largest integer less than or equal to $x$. Let $f$ be a real valued function defined on the interval $[-10,10]$ by …
  26. The value of $\int_0^{2\pi}\frac{x\sin^8x}{\sin^8x+\cos^8x}dx$ is equal to :
  27. If $[\cdot]$ represents the greatest integer function, then the value of $\left|\int_{0}^{\sqrt{\frac{\pi}{2}}}\left[\left[x^{2}\right]-\cos x\right]dx\right|$ is …
  28. $\int_{-\pi}^{\pi}\big|\pi-|x|\big|\,dx$ is equal to :
  29. Let $[t]$ denote the greatest integer less than or equal to $t$. Then, the value of the integral $\int_{0}^{1}\left[-8x^{2}+6x-1\right]dx$ is equal to
  30. The integral $\int_1^2e^x\cdot x^x(2+\log_ex)\,dx$ equals :
  31. $\int_0^2\left(\left|2x^2-3x\right|+\left[x-\frac12\right]\right)dx$, where $[t]$ is the greatest integer function, is equal to :
  32. The integral $\int_0^{\pi/2}\frac{1}{3+2\sin x+\cos x}dx$ is equal to :
  33. The value of the integral $\int_{0}^{\pi}|\sin2x|\,dx$ is $\_\_\_\_\_\_$ .
  34. If $I_{m,n}=\int_{0}^{1}x^{m-1}(1-x)^{n-1}dx$, for $m,n\ge1$, and $\int_{0}^{1}\frac{x^{m-1}+x^{n-1}}{(1+x)^{m+n}}dx=\alpha I_{m,n}$, $\alpha\in\mathbf{R}$, then …
  35. Let $[t]$ denote the greatest integer less than or equal to $t$. Then the value of the integral $\int_{-3}^{101}\left([\sin(\pi x)]+e^{[\cos(2\pi x)]}\right)dx$ is equal …
  36. Let $f$ be a real valued continuous function on $[0,1]$ and $f(x)=x+\int_{0}^{1}(x-t)f(t)\,dt$. Then, which of the following points $(x,y)$ lies on the curve $y=f(x)$?
  37. Let $I_n=\int_{1}^{e}x^{19}(\log|x|)^{n}dx$, where $n\in\mathbf{N}$. If $(20)I_{10}=\alpha I_9+\beta I_8$, for natural numbers $\alpha$ and $\beta$, then $\alpha-\beta$ …
  38. $\int_0^{20\pi}(|\sin x|+|\cos x|)^2dx$ is equal to
  39. If $\int_{-a}^{a}\left(|x|+|x-2|\right)dx=22$, $(a>2)$ and $[x]$ denotes the greatest integer $\le x$, then $\int_{a}^{-a}\left(x+[x]\right)dx$ is equal to …
  40. If $a=\lim_{n\to\infty}\sum_{k=1}^{n}\frac{2n}{n^2+k^2}$ and $f(x)=\sqrt{\frac{1-\cos x}{1+\cos x}}$, $x\in(0,1)$, then :
  41. If the value of the integral $\int_0^{1/2}\frac{x^2}{(1-x^2)^{3/2}}dx$ is $\frac k6$, then $k$ is equal to :
  42. If $[t]$ denotes the greatest integer $\le t$, then the value of $\int_0^1\left[2x-\left|3x^2-5x+2\right|+1\right]dx$ is :
  43. $\int_{0}^{5}\cos\left(\pi\left(x-\left[\frac{x}{2}\right]\right)\right)dx$, where $[t]$ denotes greatest integer less than or equal to $t$, is equal to :
  44. If $f(\alpha)=\int_1^\alpha\frac{\log_{10}t}{1+t}dt$, $\alpha>0$, then $f(e^3)+f(e^{-3})$ is equal to :
  45. $\lim_{n\to\infty}\sum_{k=1}^{n}\frac{n(n^2+k^2)}{(n^4+n^2k^2+k^4)}$ is equal to
  46. Let $I_n(x)=\int_0^x\frac{1}{(t^2+5)^n}dt$, $n=1,2,3,\ldots$. Then :
  47. The value of the definite integral $\int_{-\pi/4}^{\pi/4}\frac{dx}{(1+e^{x\cos x})(\sin^4x+\cos^4x)}$ is equal to :
  48. The value of $\int_{-2}^{2}\left|3x^{2}-3x-6\right|dx$ is $\_\_\_\_\_\_$ .
  49. The value of the integral $\int_0^{\pi/2}60\frac{\sin(6x)}{\sin x}dx$ is equal to ________.
  50. If $[x]$ denotes the greatest integer $\le x$, then the value of $\int_{-2}^{4}\big|[x]-x\big|dx$ is $\_\_\_\_\_\_$ .

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