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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 251-300 of 392

  1. Evaluate: $\displaystyle\int_0^{\pi/2}\frac{x\sin x\cos x}{\cos^{4}x+\sin^{4}x}\,dx$.
  2. Let $\dfrac{d}{dx}F(x)=\dfrac{e^{\sin x}}{x}$, $x>0$. If $\displaystyle\int_1^{4}\frac{2e^{\sin x^{2}}}{x}\,dx=F(k)-F(1)$ then one of the possible values of $k$ is …
  3. The value of $\displaystyle\int_{2}^{3}\frac{\sqrt x}{\sqrt{5-x}+\sqrt x}\,dx$ is ______.
  4. Evaluate $\displaystyle\int_0^{1}\log\left[\sqrt{1-x}+\sqrt{1+x}\right]dx$
  5. For $n>0$, $\displaystyle\int_0^{2\pi}\frac{x\sin^{2n}x}{\sin^{2n}x+\cos^{2n}x}\,dx=$ ______.
  6. Determine a positive integer $n\le5$, such that $$\int_0^1e^{x}(x-1)^{n}dx=16-6e$$
  7. Evaluate the definite integral: $\displaystyle\int_{-1/\sqrt3}^{1/\sqrt3}\left(\dfrac{x^{4}}{1-x^{4}}\right)\cos^{-1}\left(\dfrac{2x}{1+x^{2}}\right)dx$
  8. The integral $\displaystyle\int_0^{1.5}\left[x^{2}\right]dx$, where $[\ ]$ denotes the greatest integer function, equals ______.
  9. Evaluate $\displaystyle\int_0^{\pi}\frac{x\sin2x\,\sin\left(\frac{\pi}{2}\cos x\right)}{2x-\pi}\,dx$
  10. If $f(x)=A\sin\left(\dfrac{\pi x}{2}\right)+B$, $f'\left(\dfrac12\right)=\sqrt2$ and $\displaystyle\int_0^1f(x)\,dx=\dfrac{2A}{\pi}$, then constants $A$ and $B$ are
  11. The value of the integral $\displaystyle\int_{0}^{1}\sqrt{\frac{1-x}{1+x}}\,dx$ is
  12. If $y(x)=\displaystyle\int_{\pi^{2}/16}^{x^{2}}\frac{\cos x\cos\sqrt\theta}{1+\sin^{2}\sqrt\theta}\,d\theta$, then find $\dfrac{dy}{dx}$ at $x=\pi$.
  13. Find the value of $\displaystyle\int_{-\pi/3}^{\pi/3}\frac{\pi+4x^{3}}{2-\cos\left(|x|+\frac{\pi}{3}\right)}\,dx$.
  14. $\displaystyle\int_{-2}^{0}\left\{x^{3}+3x^{2}+3x+3+(x+1)\cos(x+1)\right\}dx$ is equal to
  15. The integral $\displaystyle\int_{-1/2}^{1/2}\left([x]+\ln\left(\dfrac{1+x}{1-x}\right)\right)dx$ equals
  16. Prove that $\displaystyle\int_0^{1}\tan^{-1}\left(\frac{1}{1-x+x^{2}}\right)dx=2\int_0^{1}\tan^{-1}x\,dx$. Hence or otherwise, evaluate the integral …
  17. Determine the value of $\displaystyle\int_{-\pi}^{\pi}\frac{2x(1+\sin x)}{1+\cos^{2}x}\,dx$.
  18. Let $f(x)=x-[x]$, for every real number $x$, where $[x]$ is the integral part of $x$. Then $\displaystyle\int_{-1}^{1}f(x)\,dx$ is
  19. If $\displaystyle\int_0^{x}f(t)dt=x+\int_x^{1}t\,f(t)\,dt$, then the value of $f(1)$ is
  20. Let $f(x)=\displaystyle\int_1^{x}\sqrt{2-t^{2}}\,dt$. Then the real roots of the equation $x^{2}-f'(x)=0$ are
  21. If $I(m,n)=\displaystyle\int_0^1t^m(1+t)^n\,dt$, then the expression for $I(m,n)$ in terms of $I(m+1,n-1)$ is
  22. If $g(x)=\displaystyle\int_0^{x}\cos^{4}t\,dt$, then $g(x+\pi)$ equals
  23. The value of the integral $\displaystyle\int_{e^{-1}}^{e^{2}}\left|\frac{\log_e x}{x}\right|dx$ is
  24. Let $T>0$ be a fixed real number. Suppose $f$ is a continuous function such that for all $x\in R$, $f(x+T)=f(x)$. If $I=\displaystyle\int_0^{T}f(x)\,dx$ then the value …
  25. For $x>0$, let $f(x)=\displaystyle\int_{1}^{x}\frac{\ln t}{1+t}\,dt$. Find the function $f(x)+f\left(\dfrac1x\right)$ and show that …
  26. If $f(x)=\begin{cases}e^{\cos x}\sin x, & \text{for } |x|\le2\\ 2, & \text{otherwise}\end{cases}$ then $\displaystyle\int_{-2}^{3}f(x)\,dx=$
  27. Evaluate $\displaystyle\int_{0}^{\pi}e^{|\cos x|}\left(2\sin\left(\tfrac12\cos x\right)+3\cos\left(\tfrac12\cos x\right)\right)\sin x\,dx$.
  28. $\displaystyle\int_{\pi/4}^{3\pi/4}\dfrac{dx}{1+\cos x}$ is equal to
  29. Integrate $\displaystyle\int_{0}^{\pi}\dfrac{e^{\cos x}}{e^{\cos x}+e^{-\cos x}}\,dx$.
  30. The value of $\displaystyle\int_{-\pi}^{\pi}\frac{\cos^{2}x}{1+a^{x}}\,dx$, $a>0$, is
  31. Let $g(x)=\displaystyle\int_{0}^{x}f(t)\,dt$, where $f$ is such that $\dfrac12\le f(t)\le1$ for $t\in[0,1]$ and $0\le f(t)\le\dfrac12$ for $t\in[1,2]$. Then $g(2)$ …
  32. If for a real number $y$, $[y]$ is the greatest integer less than or equal to $y$, then the value of the integral $\displaystyle\int_{\pi/2}^{3\pi/2}[2\sin x]\,dx$ is
  33. If $f(x)$ is differentiable and $\displaystyle\int_{0}^{t^{2}}xf(x)\,dx=\dfrac25t^{5}$, then $f\left(\dfrac{4}{25}\right)$ equals
  34. If $\displaystyle\int_{\sin x}^{1}t^{2}f(t)\,dt=1-\sin x$, then $f\left(\dfrac{1}{\sqrt3}\right)$ is
  35. Let $f:(0,\infty)\to R$ and $F(x)=\displaystyle\int_0^{x}f(t)\,dt$. If $F(x^{2})=x^{2}(1+x)$, then $f(4)$ equals
  36. Suppose we define the definite integral using the formula $\int_a^b f(x)dx=\dfrac{b-a}{2}\left(f(a)+f(b)\right)$, for more accurate result for $c\in(a,b)$ …
  37. Suppose we define the definite integral using the formula $\int_a^b f(x)dx=\dfrac{b-a}{2}\left(f(a)+f(b)\right)$, for more accurate result for $c\in(a,b)$ …
  38. The value of $5050\dfrac{\int_0^1(1-x^{50})^{100}dx}{\int_0^1(1-x^{50})^{101}dx}$ is
  39. If $\int_0^1 4\cot^{-1}\left(1-2x+4x^2\right)dx=a\tan^{-1}(2)-b\log_e(5)$, where $a,b\in\mathbf{N}$, then $(2a+b)$ is equal to $\_\_\_\_\_\_$ .
  40. Let $[\cdot]$ be the greatest integer function. If $\alpha=\int_0^{64}\left(x^{1/3}-\left[x^{1/3}\right]\right)dx$, then …
  41. $6\int_0^{\pi}\left|\sin3x+\sin2x+\sin x\right|dx$ is equal to $\_\_\_\_\_\_$ .
  42. The number of elements in the set $S=\{x:x\in[0,100]$ and $\int_0^xt^2\sin(x-t)\,dt=x^2\}$ is $\_\_\_\_\_\_$ .
  43. The value of $\sum_{r=1}^{20}\left(\left|\sqrt{\pi\left(\int_0^rx\left|\sin\pi x\right|dx\right)}\right|\right)$ is $\_\_\_\_\_\_$ .
  44. Let $[\,]$ denote the greatest integer function. Then $\int_{-\pi/2}^{\pi/2}\left(\frac{12(3+[x])}{3+[\sin x]+[\cos x]}\right)dx$ is equal to :
  45. Let a differentiable function $f$ satisfy the equation $\int_0^{36}f\left(\frac{tx}{36}\right)dt=4\alpha f(x)$. If $y=f(x)$ is a standard parabola passing through the …
  46. If $f(x)$ satisfies the relation $f(x)=e^x+\int_0^1\left(y+xe^x\right)f(y)\,dy$, then $e+f(0)$ is equal to $\_\_\_\_\_\_$ .
  47. The value of the integral $\int_{0}^{\infty}\frac{\log_{e}(x)}{x^{2}+4}\,dx$ is :
  48. The value of the integral $\int_{-\pi/4}^{\pi/4}\left(\frac{32\cos^{4}x}{1+e^{\sin x}}\right)dx$ is :
  49. The value of the integral $\int_{\pi/6}^{\pi/3}\left(\frac{4-\operatorname{cosec}^{2}x}{\cos^{4}x}\right)dx$ is :
  50. Let $\int_{-2}^{2}\left(|\sin x|+[x\sin x]\right)dx=2(3-\cos2)+\beta$, where $[\cdot]$ is the greatest integer function. Then $\beta\sin\left(\frac{\beta}{2}\right)$ …

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