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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 151-200 of 392

  1. Let $[\cdot]$ denote the greatest integer function. Then the value of $\int_{0}^{3}\left(\frac{e^{x}+e^{-x}}{[x]!}\right)dx$ is :
  2. The value of $\int_{1}^{a}[x]\,f'(x)\,dx$, $a>1$ where $[x]$ denotes the greatest integer not exceeding $x$ is
  3. The value of integral $\int_{3}^{6}\frac{\sqrt x}{\sqrt{9-x}+\sqrt x}\,dx$ is
  4. The value of $\int_{-2}^{3}\left|1-x^{2}\right|dx$ is
  5. The integral $\int_{2}^{4}\frac{\log x^{2}}{\log x^{2}+\log\left(36-12x+x^{2}\right)}\,dx$ is equal to :
  6. The integral $\int_{0}^{\frac12}\frac{\ln(1+2x)}{1+4x^{2}}dx$ equals
  7. Let $f(x)$ be a function satisfying $f'(x)=f(x)$ with $f(0)=1$ and $g(x)$ be a function that satisfies $f(x)+g(x)=x^{2}$. Then the value of the integral …
  8. $I_n=\int_{0}^{\pi/4}\tan^{n}x\,dx$ then $\lim_{n\to\infty}n\left[I_n+I_{n+2}\right]$ equals
  9. $\int_{0}^{2}\left[x^{2}\right]dx$ is
  10. The value of the integral $I=\int_{0}^{1}x(1-x)^{n}\,dx$ is
  11. $\int_{0}^{10\pi}|\sin x|\,dx$ is
  12. If $y=f(x)$ makes $+ve$ intercept of $2$ and $0$ unit on $x$ and $y$ axes and encloses an area of $3/4$ square unit with the axes then $\int_{0}^{2}xf'(x)\,dx$ is
  13. $\int_{-\pi}^{\pi}\frac{2x(1+\sin x)}{1+\cos^{2}x}\,dx$ is
  14. If $f(a+b-x)=f(x)$ then $\int_{a}^{b}xf(x)\,dx$ is equal to
  15. The value of the integral $\int_{1}^{2}\left(\frac{t^{4}+1}{t^{6}+1}\right)dt$ is :
  16. Let $f:\mathbb R\to\mathbb R$ be a function such that $f(2-x)=f(2+x)$ and $f(4-x)=f(4+x)$, for all $x\in\mathbb R$ and $\int_0^2f(x)\,dx=5$. Then the value of …
  17. The value of the integral $\int_{1/2}^{2}\frac{\tan^{-1}x}{x}dx$ is equal to
  18. Let $[x]$ denote the greatest integer $\le x$. Consider the function $f(x)=\max\left\{x^{2},1+[x]\right\}$. Then the value of the integral $\int_{0}^{2}f(x)\,dx$ is :
  19. For $0<a<1$, the value of the integral $\int_0^\pi\frac{dx}{1-2a\cos x+a^2}$ is :
  20. The value of the integral $\int_{-\frac\pi2}^{\frac\pi2}\sin^{4}x\left(1+\log\left(\frac{2+\sin x}{2-\sin x}\right)\right)dx$ is
  21. The value of $\int_{-\pi/2}^{\pi/2}\frac{\sin^2x}{1+2^x}dx$ is
  22. The integral $16\int_1^2\frac{dx}{x^3\left(x^2+2\right)^2}$ is equal to :
  23. Let $f:(-1,1)\to\mathbb R$ be a continuous function. If $\int_0^{\sin x}f(t)\,dt=\frac{\sqrt3}{2}x$, then $f\left(\frac{\sqrt3}{2}\right)$ is equal to :
  24. The value of $\alpha$ for which $4\alpha\int_{-1}^{2}e^{-\alpha|x|}dx=5$, is :
  25. The minimum value of the function $f(x)=\int_{0}^{2}e^{|x-t|}dt$ is
  26. Let $a$ be a positive real number such that $\int_{0}^{a}e^{x-[x]}dx=10e-9$ where $[x]$ is the greatest integer less than or equal to $x$. Then, $a$ is equal to :
  27. If $\theta_1$ and $\theta_2$ be respectively the smallest and the largest values of $\theta$ in $(0,2\pi)-\{\pi\}$ which satisfy the equation …
  28. The value of the integral $\int_{-1}^{1}\log_e\left(\sqrt{1-x}+\sqrt{1+x}\right)dx$ is equal to :
  29. The value of the integral $\int_{-2}^{2}\frac{\sin^{2}x}{\left[\frac x\pi\right]+\frac12}dx$ (where $[x]$ denotes the greatest integer less than or equal to $x$) is :
  30. The function $f(x)$ that satisfies the condition $f(x)=x+\int_{0}^{\pi/2}\sin x\cos y\,f(y)\,dy$, is :
  31. The value of the integral $\int_{4}^{10}\frac{\left[x^{2}\right]}{\left[x^{2}-28x+196\right]+\left[x^{2}\right]}dx$, where $[x]$ denotes the greatest integer less than …
  32. The value of $\int_{-\pi/2}^{\pi/2}\frac{dx}{[x]+[\sin x]+4}$, where $[t]$ denotes the greatest integer less than or equal to $t$, is :
  33. If $f(x)=\frac{2-x\cos x}{2+x\cos x}$ and $g(x)=\log_e x$, then the value of the integral $\int_{-\pi/4}^{\pi/4}g(f(x))\,dx$ is
  34. If $\int_0^{\pi/2}\frac{\cot x}{\cot x+\operatorname{cosec}x}dx=m(\pi+n)$, then $mn$ is equal to
  35. The integral $\int_{\pi/12}^{\pi/4}\frac{8\cos2x}{(\tan x+\cot x)^{3}}dx$ equals :
  36. If $I_1=\int_0^1e^{-x}\cos^{2}x\,dx$; $I_2=\int_0^1e^{-x^{2}}\cos^{2}x\,dx$ and $I_3=\int_0^1e^{-x^{3}}dx$; then
  37. Let the function $F$ be defined as $F(x)=\int_{1}^{x}\frac{e^{t}}{t}dt$, $x>0$, then the value of the integral $\int_{1}^{x}\frac{e^{t}}{t+a}dt$, where $a>0$, is
  38. The value of the integral $\int_{0}^{1}x\cot^{-1}\left(1-x^{2}+x^{4}\right)dx$ is
  39. The value of $\int_0^{2\pi}[\sin2x(1+\cos3x)]\,dx$, where $[t]$ denotes the greatest integer function is
  40. Let $f:R\to R$ be a continuous and differentiable function such that $f(2)=6$ and $f'(2)=\frac1{48}$. If $\int_6^{f(x)}4t^3dt=(x-2)g(x)$, then $\lim_{x\to2}g(x)$ is …
  41. If $\int_{1}^{2}\frac{dx}{\left(x^{2}-2x+4\right)^{3/2}}=\frac{k}{k+5}$, then $k$ is equal to
  42. If for a continuous function $f(x)$, $\int_{-\pi}^{t}\left(f(x)+x\right)dx=\pi^{2}-t^{2}$, for all $t\ge-\pi$, then $f\left(-\frac\pi3\right)$ is equal to :
  43. Let $I=\int_{a}^{b}\left(x^{4}-2x^{2}\right)dx$. If $I$ is minimum then the ordered pair $(a,b)$ is
  44. The value of $\int_{0}^{\pi/2}\frac{\sin^{3}x}{\sin x+\cos x}dx$ is :
  45. The integral $\int_{\pi/4}^{3\pi/4}\frac{dx}{1+\cos x}$ is equal to :
  46. If $\int_0^{\pi/3}\frac{\tan\theta}{\sqrt{2k\sec\theta}}d\theta=1-\frac1{\sqrt2}$, $(k>0)$, then the value of $k$ is
  47. The integral $\int_{\pi/6}^{\pi/4}\frac{dx}{\sin2x\left(\tan^{5}x+\cot^{5}x\right)}$ equals :
  48. If for $n\ge1$, $P_n=\int_1^e(\log x)^n\,dx$, then $P_{10}-90P_8$ is equal to :
  49. If $\int_0^xf(t)\,dt=x^2+\int_x^1t^2f(t)\,dt$, then $f'\left(\frac12\right)$ is :
  50. If $[\ ]$ denotes the greatest integer function, then the integral $\int_{0}^{\pi}[\cos x]\,dx$ is equal to :

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