Integral Calculus Questions - JEE

Consider curves $C_1: y^2-x=0$; $C_2: y-x^2=0$; $0\leq x\leq\frac{\sqrt{3}}{2}$ and $C_3: y=f(x)$; $f(x)<0$ $\forall x\in\left(0,\frac{\sqrt{3}}{2}\right)$. From any point $P$ on $C_2$, lines parallel to coordinate axes intersect $C_1$ at $Q$ and $C_3$ at $R$. If area of region $OPRO$ is twice the area of region $OPQO$ (O = origin), then $\left|32f\!\left(\frac{1}{2}\right)\right|$ is

Let a function $f(x)$ be defined in $[-2, 2]$ as $f(x) = \begin{cases} \{x\}, & -2 \leq x < -1 \\ |\text{sgn } x|, & -1 \leq x \leq 1 \\ \{-x\}, & 1 < x \leq 2 \end{cases}$, where $\{x\}$ denotes fractional part, then area bounded by graph of $f(x)$ and $x$-axis is:

A strictly increasing continuous function $f(x)$ intersects its inverse $f^{-1}(x)$ at $x=\alpha$ and $x=\beta$, $\displaystyle\int_\alpha^\beta (f(x)+f^{-1}(x))\,dx=13$, where $\alpha,\beta\in\mathbb{N}$. Then $|\alpha\beta|$ equals

Let $f:[-1,0] \to R$ be a function differentiable within the domain and that $\displaystyle\int_{-1}^{0}(f(x))^2\,dx = 10$ and $f(-1) = 2$. The value of the integral $\displaystyle\int_{-1}^{0} x f'(x) f(x)\,dx$, is:

If $f(x)$ and $g(x)$ are both continuous functions then the value of $\displaystyle\int_{\ln\lambda}^{\ln(1/\lambda)} \dfrac{f\!\left(\dfrac{x^2}{4}\right)(f(x) - f(-x))}{g\!\left(\dfrac{x^2}{4}\right)(g(x) + g(-x))}\,dx$ is equal to:

If $\displaystyle\int (x^2+1)(x+1)^2 e^x\,dx = A(f(x))^2 + C$ and $f(-1) = \frac{2}{e}$, then $2A + f(0)$ equals

Let $I(x)=\displaystyle\int\frac{x+1}{x(x^2e^{2x}-1)}\,dx=\frac{1}{4}\ln\frac{(xe^x)^\alpha-2(xe^x)^\beta+\gamma}{x^4e^{4x}}+c$, then $\alpha+\beta+\gamma$ equals

The area bounded by the $x$-axis, part of the curve $y=1+x^{-2}$ and the ordinates $x=1$, $x=2$ is divided into equal parts by the ordinate at $x=a$ such that $a=\dfrac{3+\sqrt{p}}{8}$. The value of $p$ is

If $\displaystyle\int e^{\frac{1}{2}\left(x^2+\frac{1}{x^2}\right)}\cdot\frac{x^4+x^2-1}{x^2}\,dx = f(x)+c$, then $\bigl(f(\sqrt{2})\bigr)^4$ is

Value of $\displaystyle\int_0^1 x^6(x^3-1)^{2022}\,dx$ is

For positive integer $n$, let $I_n=\displaystyle\int_{-\pi}^{\pi}\!\left(\frac{\pi}{2}-|x|\right)\cos nx\,dx$. Find $[I_1+I_2+I_3+I_4]$ (GIF).

The area enclosed between the curves $y=ax^2$ and $x=ay^2$ ($a>0$) is 1 sq. unit. Then the value of $a$ is

Area enclosed by $y=g(x)$, $x=1$ and $x=37$, where $g(x)$ is the inverse of $f(x)=x^3+3x+1$, is

If $f(x) = \sin x + \displaystyle\int_{-\pi/2}^{\pi/2}(\sin x + t\cos x)f(t)\,dt$, then $f(x)$ may be equal to $\left(-\dfrac{1}{k}\sin x - \dfrac{2}{k}\cos x\right)$, where $k$ is a numerical quantity which equals

Integral Calculus Questions (265)