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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 301-350 of 392

  1. Let $f$ be a real polynomial of degree $n$ such that $f(x)=f'(x)f''(x)$, for all $x\in\mathbb{R}$. If $f(0)=0$, then $36\left(f'(2)+f''(2)+\int_{0}^{2}f(x)\,dx\right)$ …
  2. For any real number $x$, let $[x]$ denote the largest integer less than or equal to $x$. If $$I=\int_0^{10}\left[\sqrt{\frac{10x}{x+1}}\right]dx,$$ then the value of …
  3. Let $f:\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\to\mathbb{R}$ be a continuous function such that $$f(0)=1 \text{ and } \int_0^{\pi/3} f(t)\,dt = 0$$ Then which of the …
  4. The value of $\int_{-\pi/2}^{\pi/2}\left(\frac{1}{[x]+4}\right)dx$, where $[\cdot]$ denotes the greatest integer function, is :
  5. The value of the integral $\int_{\pi/24}^{5\pi/24}\frac{dx}{1+\sqrt[3]{\tan2x}}$ is :
  6. The value of $\int_{-\pi/6}^{\pi/6}\left(\frac{\pi+4x^{11}}{1-\sin\left(|x|+\frac{\pi}{6}\right)}\right)dx$ is equal to :
  7. Let $f$ be a polynomial function such that $f(x^2+1)=x^4+5x^2+2$, for all $x\in\mathbf{R}$. Then $\int_0^3f(x)\,dx$ is equal to :
  8. State TRUE or FALSE: The value of the integral $\displaystyle\int_0^{2a}\left[\dfrac{f(x)}{\{f(x)+f(2a-x)\}}\right]dx$ is equal to $a$.
  9. Let $S_n=\sum_{k=1}^{n}\dfrac{n}{n^2+kn+k^2}$ and $T_n=\sum_{k=0}^{n-1}\dfrac{n}{n^2+kn+k^2}$, for $n=1,2,3,\ldots$ Then,
  10. Let $f:[0,2]\to\mathbb{R}$ be a function which is continuous on $[0,2]$ and is differentiable on $(0,2)$ with $f(0)=1$. Let $$F(x)=\int_0^{x^2}f(\sqrt t)\,dt$$ for …
  11. The following integral $$\int_{\pi/4}^{\pi/2}(2\operatorname{cosec}x)^{17}dx$$ is equal to
  12. The value of $$\int_0^1 4x^3\left\{\frac{d^2}{dx^2}(1-x^2)^5\right\}dx$$ is
  13. The value of $\int_{-\pi/2}^{\pi/2}\frac{x^2\cos x}{1+e^x}dx$ is equal to
  14. Let $a,b,c$ be non-zero real numbers such that …
  15. Let $f(x)=x+\frac{a}{\pi^{2}-4}\sin x+\frac{b}{\pi^{2}-4}\cos x$, $x\in R$ be a function which satisfies $f(x)=x+\int_{0}^{\pi/2}\sin(x+y)f(y)\,dy$. Then $(a+b)$ is …
  16. For $x>0$, let $f(x)=\int_1^x\frac{\log t}{1+t}dt$. Then $f(x)+f\left(\frac1x\right)$ is equal to
  17. Let $f$ be a non-negative function defined on the interval $[0,1]$. If $$\int_0^x\sqrt{1-(f'(t))^2}\,dt=\int_0^x f(t)\,dt,\quad 0\le x\le 1,$$ and $f(0)=0$, then
  18. If $$I_n=\int_{-\pi}^{\pi}\frac{\sin nx}{(1+\pi^x)\sin x}\,dx,\quad n=0,1,2,\ldots,$$ then
  19. Let $f:\mathbf{R}\to\mathbf{R}$ be a continuous function which satisfies $$f(x)=\int_0^x f(t)\,dt.$$ Then the value of $f(\ln 5)$ is
  20. 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 …
  21. The value(s) of $\int_0^1\frac{x^4(1-x)^4}{1+x^2}\,dx$ is (are)
  22. The value of $\int_{\sqrt{\ln 2}}^{\sqrt{\ln 3}}\frac{x\sin x^2}{\sin x^2+\sin(\ln 6-x^2)}dx$ is
  23. Let $f:[-1,2]\to[0,\infty)$ be a continuous function such that $f(x)=f(1-x)$ for all $x\in[-1,2]$. Let $R_1=\int_{-1}^{2}x f(x)\,dx$, and $R_2$ be the area of the region …
  24. The value of the integral $$\int_{-\pi/2}^{\pi/2} \left(x^2 + \ln\frac{\pi + x}{\pi - x}\right)\cos x\,dx$$ is
  25. Let $f:\left[\frac{1}{2},1\right]\to\mathbb{R}$ (the set of all real numbers) be a positive, non-constant and differentiable function such that $f'(x)<2f(x)$ and …
  26. Let $F(x)=\int_{x}^{x^2+\frac{\pi}{6}}2\cos^2 t\,dt$ for all $x\in\mathbb{R}$ and $f:\left[0,\frac12\right]\to[0,\infty)$ be a continuous function. For …
  27. Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined by $f(x)=\begin{cases}[x], & x\le 2\\ 0, & x>2\end{cases}$, where $[x]$ is the greatest integer less than or equal …
  28. $$\begin{array}{ll}\textbf{List I} & \textbf{List II}\\ \text{P. The number of polynomials } f(x) \text{ with non-negative integer coefficients of degree} \le 2 \text …
  29. Let $f'(x)=\frac{192x^3}{2+\sin^4\pi x}$ for all $x\in\mathbb{R}$ with $f\left(\frac12\right)=0$. If $m\le\int_{1/2}^{1}f(x)\,dx\le M$, then the possible values of $m$ …
  30. If $$\alpha=\int_0^1\left(e^{9x+3\tan^{-1}x}\right)\left(\frac{12+9x^2}{1+x^2}\right)dx$$ where $\tan^{-1}x$ takes only principal values, then the value of …
  31. Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous odd function, which vanishes exactly at one point and $f(1)=\frac{1}{2}$. Suppose that $F(x)=\int_{-1}^{x}f(t)\,dt$ for …
  32. Let $f(x)=7\tan^8x+7\tan^6x-3\tan^4x-3\tan^2x$ for all $x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. Then the correct expression(s) is(are)
  33. The option(s) with the values of $a$ and $L$ that satisfy the following equation is(are) …
  34. The total number of distinct $x\in[0,1]$ for which $\int_0^x\frac{t^2}{1+t^4}dt=2x-1$ is
  35. Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that $f(0)=0$, $f\left(\frac{\pi}{2}\right)=3$ and $f'(0)=1$. If $$g(x)=\int_x^{\pi/2}\left[f'(t)\csc …
  36. Which of the following inequalities is/are TRUE?
  37. The value of the integral $$\int_0^{1/2}\frac{1+\sqrt{3}}{\left((x+1)^2(1-x)^6\right)^{1/4}}\,dx$$ is _____ .
  38. Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that its derivative $f'$ is continuous and $f(\pi)=-6$. If $F:[0,\pi]\to\mathbb{R}$ is defined by …
  39. If $\int_{-\pi/2}^{\pi/2}\frac{96x^2\cos^2 x}{(1+e^x)}\,dx=\pi(\alpha\pi^2+\beta)$, $\alpha,\beta\in\mathbb{Z}$, then $(\alpha+\beta)^2$ equals
  40. The value of $2\int_0^{\pi/2}f(x)g(x)\,dx-\int_0^{\pi/2}g(x)\,dx$ is __________.
  41. The value of $\frac{16}{\pi^3}\int_0^{\pi/2}f(x)g(x)\,dx$ is ____________.
  42. If $\int_1^3x^2F'(x)\,dx=-12$ and $\int_1^3x^3F''(x)\,dx=40$, then the correct expression(s) is(are)
  43. $\displaystyle\int_{-1}^{1}g'(x)\,dx=$
  44. The value of $g\left(\frac12\right)$ is
  45. Which of the following statements is TRUE ?
  46. The value of $\frac{16S_1}{\pi}$ is ___ .
  47. Which of the following statements is TRUE ?
  48. The value of $\frac{48S_2}{\pi^2}$ is ___.
  49. The value of $g'\left(\frac12\right)$ is
  50. Let $p(x)$ be a function defined on $R$ such that $p'(x)=p'(1-x)$, for all $x\in[0,1]$, $p(0)=1$ and $p(1)=41$. Then $\int_{0}^{1}p(x)\,dx$ equals

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