Definite Integration Questions - JEE

The value of $\displaystyle\int_{-\pi/2}^{\pi/2} \frac{\sin^2 x}{1+2^x}\, dx$ is

If $\int f(x)dx = \Psi(x)$, then $\int x^5 f(x^3)\,dx$ is equal to

The integral $\displaystyle\int_2^4 \dfrac{\log x^2}{\log x^2 + \log(36-12x+x^2)}\,dx$ is equal to

The average value of the electromotive force $E_m$ over the duration $t \in [0, T]$ where the electromotive force $E$ is computed by $E = E_0 \sin\frac{2\pi t}{T}$, $E_0$ and $T$ being constants, is equal to

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

The value of $\displaystyle\lim_{n\to\infty}\left(\ln\!\left(\sqrt[n]{\frac{4}{n^2}}\right)+\ln\!\left(\sqrt[n]{\frac{16}{n^2}}\right)+\ln\!\left(\sqrt[n]{\frac{36}{n^2}}\right)+\cdots+\ln\!\left(\sqrt[n]{\frac{4n^2}{n^2}}\right)\right)$ equals:

The value of \[ \lim_{n \to \infty} \sum_{r=1}^{n} \frac{\pi}{n} \cdot \frac{1}{\sin\left( \dfrac{(n+r)\pi}{4n} \right)} \] is equal to:

The value of $\int_2^3 \frac{dz}{z^2 - 1}$ is

If $y = \int_0^{f(x)} \frac{dt}{1+t^4}$, find $\frac{dy}{dx}$.

If p(x) is a polynomial such that p'(x) = p'(1 − x) and p(0) = 42, find the value of I = $\int_0^1 p(x)\,dx$.

Let \[I = \int \frac{dx}{(x^2 - 2x + 10)^2}\]If $I = A\left[\tan^{-1}\left(\frac{x-1}{3}\right) + \frac{f(x)}{x^2 - 2x + 10}\right] + C$, then $A$ and $f(x)$ are:

Let $f(x) = \begin{vmatrix} 2\cos^2 x & \sin 2x & -\sin x \\ \sin 2x & 2\sin^2 x & \cos x \\ \sin x & -\cos x & 0 \end{vmatrix}$. Then the value of $\displaystyle\int_0^{\pi/2} [f(x) + f'(x)]\,dx$ is

If $\int x^5 e^{-x^2} dx = g(x)e^{-x^2} + C$, where $C$ is a constant of integration, then $g(-1)$ is equal to:

Evaluate $\int \frac{1+x}{\sqrt{1+x^4+x^2}} \cdot \frac{1}{1+x^4} dx$

$\int_0^{\pi/2} \left(\int_0^{x^2} \cos x\, dx\right)\sin y\, dy =$ ______.

The integral $\displaystyle\int_{\pi/6}^{\pi/3} \sec^{2/3} x\,\csc^{4/3} x\,dx$ is equal to:

Given $\int x^5 e^{-4x^3} dx = \frac{1}{48} e^{-4x^3} f(x) + C$Then $f(x)$ is equal to:

Evaluate $\int_{-20\pi}^{20\pi} |\cos x|\,dx$.

Evaluate $\int \frac{dx}{9-16x^2}$

Let f(x) be a continuous function in (0, 1) satisfying $\int_0^1 x\sqrt{x}\, f(x)(1 - \sqrt{x}\,f(x))\,dx = \dfrac{1}{8}$. Number of solutions of the equation $f(x) = e^x$ is:

The value of $\displaystyle\int_0^{\pi/2} \frac{\sin^3 x}{\sin x + \cos x}\,dx$ is:

Suppose that a continuous function $f(x)$ satisfies the relation $\int_{x}^{x+1} f(t)\, dt = e^x$ for every $x \geq 0$. The value of $f(2) - f(0)$, equals:

If a, b, c be nonzero real numbers such that$\int_0^1 (1+\cos^8 x)(ax^2+bx+c)dx = \int_0^2 (1+\cos^8 x)(ax^2+bx+c)dx$then the quadratic equation $ax^2+bx+c=0$ has

Evaluate the integral: $\int_{-1}^{2} x dx$

Let \[ m = \int_{-2}^{0} \frac{|\sin x|}{\left[\dfrac{x}{\pi}\right] + \dfrac{1}{2}}\, dx \] and \[ n = \int_{0}^{2} \frac{|\sin t|}{-\left[\dfrac{t}{\pi}\right] - 1 + \dfrac{1}{2}}\, dt \] Find the value of $-\dfrac{m}{n}$.

The value of $\dfrac{\displaystyle\int_0^{\pi/2}(5\cos^2 x + 3\sin^2 x)\,dx}{\displaystyle\int_0^{\pi/2}\sin\theta\cos\theta\sqrt{25\sin^2\theta + 9\cos^2\theta}\,d\theta}$ is equal to:

If for a continuous function $f(x)$, $\int_{-\pi}^{t} (f(x) + x) \, dx = \pi^2 - t^2$, for all $t \geq -\pi$, then $f\!\left(-\dfrac{\pi}{3}\right)$ is equal to

Find $\int \frac{dx}{x^2 - a^2}$

Evaluate the integral $I = \int_{2}^{e} \left( \frac{1}{\ln x} - \frac{1}{\ln^2 x} \right) dx$.

The evaluation of $\displaystyle\int e^{x-\frac{1}{x}}\cdot\frac{x^2+1}{x^2}\,dx$ is

Let $x = f''(t)\cos t + f'(t)\sin t$ and $y = -f''(t)\sin t + f'(t)\cos t$. Then $\int\left[\left(\dfrac{dx}{dt}\right)^2 + \left(\dfrac{dy}{dt}\right)^2\right]^{1/2} dt$ is equal to

$\int_{0}^{\pi/2} (2\log \sin x - \log \sin 2x) dx$ equals -

We have $x\displaystyle\int_0^x (1-t)f(t)\,dt = \int_0^x t\,f(t)\,dt$. If $f(1) = \alpha$, then find $\alpha$.

The value of $\displaystyle\lim_{n\to\infty}\dfrac{1+2-3+4+5-6+\cdots+(3n-2)+(3n-1)-3n}{\sqrt{2n^4+4n+3}-\sqrt{n^4+5n+4}}$ is:

If $I = \int \dfrac{1}{2\sin x \cos x}\,dx + \dfrac{1}{2}\int \dfrac{\sqrt{\tan x}}{\sin x \cos x}\,dx$, and $f(x) = \tan^2 x$, find $f\!\left(\dfrac{\pi}{3}\right)$.

If $\int_{-\pi}^{t}(f(x)+x)\,dx = \pi^2 - t^2$, then $f\!\left(\dfrac{\pi}{6}\right)$ equals:

Evaluate $\int_{0}^{\pi/2} \cos^6 x \, dx$.

If $f(x)$ is a continuous function such that $f(x) \ge 0 \ \forall \ x \in [2,10]$ and $\int_{4}^{8} f(x) dx = 0$, then find $f(6)$.

Evaluate $\int \frac{(\sin x - 2\cos x)}{(2\sin x + \cos x)} dx$

If $a > 1$, then $\int_1^a [x]\,f'(x)\,dx$ equals (where $[x]$ denotes greatest integer function):

If $\displaystyle\lim_{n\to\infty}\sum_{k=1}^{n}\frac{e^{\frac{k}{n}}+e^{\frac{-k}{n}}}{n\sqrt{1-e^{\frac{2k}{n}}-e^{\frac{-2k}{n}}}}=\sin^{-1}\!\left(\frac{e^a-e^{-a}}{b}\right)$ where $a$ and $b$ are positive integers, then the value of $a+b$ is:

The solution for $x$ of the equation $\int_{\sqrt{2}}^{x} \dfrac{dt}{t\sqrt{t^2-1}} = \dfrac{\pi}{2}$ is

Evaluate $\int_{0}^{\pi/2} \sin^4 2x \, dx$.

Show that $\int_{0}^{n\pi+1} |\sin x|\,dx = 2n + 1 - \cos\theta$, where $n \in \mathbb{N}$ and $0 \le \theta \le \pi$.

Evaluate $\int_{0}^{2\pi} \frac{\sin 2x}{\cos 4x + \sin \frac{x}{2}} \, dx$

Let f(x) and g(x) be continuous, positive functions such that $f(-x) = g(x) - 1$, $f(x) = \dfrac{g(x)}{g(-x)}$ and $\displaystyle\int_{-20}^{20} f(x)\,dx = 2020$, then the value of $\displaystyle\int_{-20}^{20} \dfrac{f(x)}{g(x)}\,dx$ is:

If the system of equations $2x - y + z = 0$, $x - 2y + z = 0$ and $ax - y + 2z = 0$ has infinitely many solutions and $f(x)$ be a continuous function such that $f(5+x) + f(x) = 2$ $\forall\, x \in R$, then $\displaystyle\int_0^{-2a} f(x)\,dx$ is equal to:

If $\int x^{26}(x-1)^{17}(5x-3)\, dx = \frac{1}{k} x^{27}(x-1)^{18} + C$, then the value of $k$ is:

Let \[ I = \int_{0}^{1} \frac{\sin t}{1+t}\, dt = \alpha \] and \[ I_1 = \int_{4\pi-2}^{4\pi} \frac{\sin(t/2)}{4\pi + 2 - t}\, dt \] If $I_1 = -k\alpha$, find the value of $k$.

The value of $\int_{0}^{2} (x^2) dx \left\{ \int_{a}^{b} f(x) dx = \int_{a}^{c} f(x)dx + \int_{c}^{b} f(x)dx \right\}$ where $f(x) = [x^2]$ (greatest integer function) is equal to:

Definite Integration Questions (1340)