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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 101-150 of 392

  1. The integral $\int_0^2\big||x-1|-x\big|\,dx$ is equal to ________.
  2. Let $[t]$ denote the greatest integer $\le t$. Then $\frac{2}{\pi}\int_{\pi/6}^{5\pi/6}\left(8[\operatorname{cosec}x]-5[\cot x]\right)dx$ is equal to $\_\_\_\_\_\_$ .
  3. Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined by $f(x)=\frac{4^{x}}{4^{x}+2}$ and $M=\int_{f(a)}^{f(1-a)}x\sin^{4}(x(1-x))\,dx$ …
  4. Let $f(x)=\min\{[x-1],[x-2],\ldots,[x-10]\}$ where $[t]$ denotes the greatest integer $\le t$. Then $\int_0^{10}f(x)\,dx+\int_0^{10}(f(x))^2dx+\int_0^{10}|f(x)|\,dx$ is …
  5. Let $[\cdot]$ denote the greatest integer function. If $\int_{0}^{e^{3}}\left[\frac{1}{e^{x-1}}\right]dx=\alpha-\log_e2$, then $\alpha^{3}$ is equal to $\_\_\_\_\_\_$ .
  6. If the integral $525\int_{0}^{\pi/2}\sin2x\cos^{\frac{11}{2}}x\left(1+\cos^{\frac{5}{2}}x\right)^{\frac{1}{2}}dx$ is equal to $\left(n\sqrt{2}-64\right)$, then $n$ is …
  7. Let $f_n=\int_{0}^{\pi/2}\left(\sum_{k=1}^{n}\sin^{k-1}x\right)\left(\sum_{k=1}^{n}(2k-1)\sin^{k-1}x\right)\cos x\,dx$, $n\in\mathbb{N}$. Then $f_{21}-f_{20}$ is equal …
  8. The value of $b>3$ for which $12\int_{3}^{b}\frac{1}{(x^{2}-1)(x^{2}-4)}dx=\log_e\left(\frac{49}{40}\right)$, is equal to $\_\_\_\_\_\_$ .
  9. Let $f$ be a differentiable function satisfying $f(x)=\frac{2}{\sqrt3}\int_0^{\sqrt3}f\left(\frac{\lambda^2x}{3}\right)d\lambda$, $x>0$ and $f(1)=\sqrt3$. If $y=f(x)$ …
  10. The value of the integral $\frac{48}{\pi^{4}}\int_{0}^{\pi}\left(\frac{3\pi x^{2}}{2}-x^{3}\right)\frac{\sin x}{1+\cos^{2}x}dx$ is equal to $\_\_\_\_\_\_$ .
  11. If $\int_0^\pi(\sin^3x)e^{-\sin^2x}dx=\alpha-\frac\beta e\int_0^1\sqrt t\,e^t\,dt$, then $\alpha+\beta$ is equal to ________.
  12. Let $r_k=\frac{\int_{0}^{1}\left(1-x^{7}\right)^{k}dx}{\int_{0}^{1}\left(1-x^{7}\right)^{k+1}dx}$, $k\in\mathbb{N}$. Then the value of …
  13. If $\int_0^{\sqrt3}\frac{15x^3}{\sqrt{1+x^2+\sqrt{(1+x^2)^3}}}dx=\alpha\sqrt2+\beta\sqrt3$, where $\alpha,\beta$ are integers, then $\alpha+\beta$ is equal to ________.
  14. Let $a_n=\int_{-1}^{n}\left(1+\frac x2+\frac{x^2}{3}+\ldots+\frac{x^{n-1}}{n}\right)dx$ for every $n\in\mathbb{N}$. Then the sum of all the elements of the set …
  15. The integral $\frac{24}{\pi}\int_{0}^{\sqrt{2}}\frac{\left(2-x^{2}\right)dx}{\left(2+x^{2}\right)\sqrt{4+x^{4}}}$ is equal to $\_\_\_\_\_\_$ .
  16. Let $f:(0,\infty)\to\mathbf{R}$ and $F(x)=\int_{0}^{x}tf(t)\,dt$. If $F(x^{2})=x^{4}+x^{5}$, then $\sum_{r=1}^{12}f(r^{2})$ is equal to $\_\_\_\_\_\_$ .
  17. Let the domain of the function $f(x)=\log_4\left(\log_5\left(\log_3(18x-x^2-77)\right)\right)$ be $(a,b)$. Then the value of the integral …
  18. Let $F:[3,5]\to\mathbf{R}$ be a twice differentiable function on $(3,5)$ such that $F(x)=e^{-x}\int_3^x(3t^2+2t+4F'(t))\,dt$. If …
  19. Let $f$ be a twice differentiable function on $\mathbb{R}$. If $f'(0)=4$ and $f(x)+\int_0^x(x-t)f'(t)\,dt=(e^{2x}+e^{-2x})\cos2x+\frac2a x$, then $(2a+1)^5a^2$ is equal …
  20. For $m,n>0$, let $\alpha(m,n)=\int_{0}^{2}t^{m}(1+3t)^{n}dt$. If $11\alpha(10,6)+18\alpha(11,5)=p(14)^{6}$, then $p$ is equal to $\_\_\_\_\_\_$ .
  21. Let $f(\theta)=\sin\theta+\int_{-\pi/2}^{\pi/2}(\sin\theta+t\cos\theta)f(t)\,dt$. Then the value of $\left|\int_{0}^{\pi/2}f(\theta)\,d\theta\right|$ is $\_\_\_\_\_\_$ .
  22. The value of $9\int_{0}^{9}\left[\sqrt{\frac{10x}{x+1}}\right]dx$, where $[t]$ denotes the greatest integer less than or equal to $t$, is $\_\_\_\_\_\_$ .
  23. Let $[t]$ denote the greatest integer function. If $\int_{0}^{2.4}\left[x^{2}\right]dx=\alpha+\beta\sqrt{2}+\gamma\sqrt{3}+\delta\sqrt{5}$, then …
  24. If $\int_{-0.15}^{0.15}\left|100x^{2}-1\right|dx=\frac{k}{3000}$, then $k$ is equal to $\_\_\_\_\_\_$ .
  25. Let $[t]$ denote the largest integer less than or equal to $t$. If …
  26. Let $\{x\}$ and $[x]$ denote the fractional part of $x$ and the greatest integer $\le x$ respectively of a real number $x$. If $\int_0^n\{x\}\,dx$, $\int_0^n[x]\,dx$ and …
  27. $\left|\frac{120}{\pi^{3}}\int_{0}^{\pi}\frac{x^{2}\sin x\cos x}{\sin^{4}x+\cos^{4}x}dx\right|$ is equal to $\_\_\_\_\_\_$ .
  28. If $\int_{-\pi/2}^{\pi/2}\frac{8\sqrt{2}\cos x\,dx}{\left(1+e^{\sin x}\right)\left(1+\sin^{4}x\right)}=\alpha\pi+\beta\log_e\left(3+2\sqrt{2}\right)$, where …
  29. Let $[t]$ denote the greatest integer $\le t$. Then the value of $8\cdot\int_{-\frac{1}{2}}^{1}\left([2x]+|x|\right)dx$ is $\_\_\_\_\_\_$ .
  30. Let $f(x)=\max\{|x+1|,|x+2|,\ldots,|x+5|\}$. Then $\int_{-6}^{0}f(x)\,dx$ is equal to $\_\_\_\_\_\_$ .
  31. Let for $x\in\mathbb{R}$, $S_0(x)=x$, $S_k(x)=C_kx+k\int_{0}^{x}S_{k-1}(t)\,dt$, where $C_0=1$, $C_k=1-\int_{0}^{1}S_{k-1}(x)\,dx$, $k=1,2,3,\ldots$. Then $S_2(3)+6C_3$ …
  32. Let $\max_{0\le x\le2}\left\{\frac{9-x^{2}}{5-x}\right\}=\alpha$ and $\min_{0\le x\le2}\left\{\frac{9-x^{2}}{5-x}\right\}=\beta$. If …
  33. If $n(2n+1)\int_0^1(1-x^n)^{2n}dx=1177\int_0^1(1-x^n)^{2n+1}dx$, then $n\in\mathbf{N}$ is equal to ________.
  34. Let $[t]$ denote the greatest integer less than or equal to $t$. Then the value of $\int_1^2|2x-[3x]|\,dx$ is ________.
  35. The value of $I=\int_{0}^{\pi/2}\frac{(\sin x+\cos x)^{2}}{\sqrt{1+\sin2x}}\,dx$ is
  36. If $\int_{0}^{\pi}xf(\sin x)\,dx=A\int_{0}^{\pi/2}f(\sin x)\,dx$, then $A$ is
  37. If $f(x)=\frac{e^{x}}{1+e^{x}}$, $I_1=\int_{f(-a)}^{f(a)}xg\{x(1-x)\}\,dx$ and $I_2=\int_{f(-a)}^{f(a)}g\{x(1-x)\}\,dx$, then the value of $\frac{I_2}{I_1}$ is
  38. The value of $\int_{-\pi/2}^{\pi/2}\frac{\sin^{2}x}{1+2^{x}}\,dx$ is :
  39. The solution for $x$ of the equation $\int_{\sqrt2}^{x}\frac{dt}{t\sqrt{t^{2}-1}}=\frac\pi2$ is
  40. Let $f:[-2,3]\to[0,\infty)$ be a continuous function such that $f(1-x)=f(x)$ for all $x\in[-2,3]$. If $R_1$ is the numerical value of the area of the region bounded by …
  41. $\int_{0}^{\pi}x\,f(\sin x)\,dx$ is equal to
  42. Statement-1 : The value of the integral $\int_{\pi/6}^{\pi/3}\frac{dx}{1+\sqrt{\tan x}}$ is equal to $\pi/6$. Statement-2 …
  43. Let $F(x)=f(x)+f\left(\frac1x\right)$, where $f(x)=\int_{1}^{x}\frac{\log t}{1+t}\,dt$. Then $F(e)$ equals
  44. Let $I=\int_{0}^{1}\frac{\sin x}{\sqrt x}\,dx$ and $J=\int_{0}^{1}\frac{\cos x}{\sqrt x}\,dx$. Then which one of the following is true?
  45. If $I_1=\int_{0}^{1}2^{x^{2}}dx$, $I_2=\int_{0}^{1}2^{x^{3}}dx$, $I_3=\int_{1}^{2}2^{x^{2}}dx$ and $I_4=\int_{1}^{2}2^{x^{3}}dx$ then
  46. For $0\le x\le\frac\pi2$, the value of $\int_{0}^{\sin^{2}x}\sin^{-1}\left(\sqrt t\right)dt+\int_{0}^{\cos^{2}x}\cos^{-1}\left(\sqrt t\right)dt$ equals :
  47. The integral $\int_{0}^{\pi}\sqrt{1+4\sin^{2}\frac{x}{2}-4\sin\frac{x}{2}}\,dx$ equals :
  48. If $f(y)=e^{y}$, $g(y)=y$; $y>0$ and $F(t)=\int_{0}^{t}f(t-y)g(y)\,dy$, then :
  49. $\int_{-3\pi/2}^{-\pi/2}\left[(x+\pi)^{3}+\cos^{2}(x+3\pi)\right]dx$ is equal to
  50. The value of $\int_{-\pi}^{\pi}\frac{\cos^{2}x}{1+a^{x}}\,dx$, $a>0$, is

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