Definite Integration Questions - JEE

Evaluate: \[I = \int_{-\pi/2}^{\pi/2} \frac{2}{1+e^x}\, dx\]

For $n > 0$, $\int_{0}^{\pi} \frac{x\sin 2nx}{\sin^{2n} x + \cos^{2n} x} dx$ is

The value of definite integral $ \displaystyle\int_{-\pi}^{\pi} \dfrac{2x(1 + \sin x)}{1 + \cos^2 x}\, dx $ is:

Given $ I = \int_{\pi/6}^{\pi/3} \sec^{2/3} x \cdot \csc^{4/3} x \, dx $, find the value of $I$.

Evaluate $\displaystyle\int_0^{\pi/2}x\cos x\,dx$ [JEE Main 2018]

Let $f(x) = \int_1^x \dfrac{\log t}{1+t}\,dt$. Then $F(e) = f(e) + f\!\left(\dfrac{1}{e}\right)$ equals:

Let $I = \int_{a}^{b} (x^4 - 2x^2) \, dx$. If $I$ is minimum then the ordered pair $(a, b)$ is:

Evaluate: $\displaystyle\int_0^{2\pi} \frac{x\sin^{2n} x}{\sin^{2n} x + \cos^{2n} x}\,dx$ (up to four decimal places).

$\displaystyle\lim_{n\to\infty}\left(\dfrac{1}{1+n}+\dfrac{1}{2+n}+\dfrac{1}{3+n}+\cdots+\dfrac{1}{2n}\right)$ is equal to:

If $a_1, a_2$ and $a_3$ are the three values of $a$ which satisfy the equation $\int_0^{\pi/2} (\sin x + a \cos x)^3 dx = \frac{4a}{\pi - 2} \int_0^{\pi/2} x \cos x dx = 2$ then $\left(a_1^2 + a_2^2 + a_3^2\right)$ is equal to ____.

$\lim\limits _{n \rightarrow \infty} \frac{3}{n}$$\left\{4+\left(2+\frac{1}{n}\right)^2+\left(2+\frac{2}{n}\right)^2+. .+\left(3-\frac{1}{n}\right)^2\right\}$ is equal

Given $ g(x) = f(x+5) $, $ f(0) = 0 $, and $ g(x) $ is an even function. Find $ I = \int_0^x f(t)\,dt $ in terms of $g$.

The integral $\displaystyle\int_{\pi/2}^{\pi/4} \dfrac{8\cos 2x}{(\tan x + \cot x)^3}\,dx$ equals

If $f(x)$ is a differentiable function defined for all positive real numbers such that $xf(x) = x + \int_{1}^{x} f(t)\, dt$, then the value of $\sum_{k=1}^{10} f(e^k)$ is:

Evaluate: $ I = \int_{\pi/4}^{3\pi/4} \dfrac{x}{1+\sin x} \, dx $(1) $\pi(\sqrt{2}+1)$ (2) $\pi(\sqrt{2}-1)$ (3) $\pi\sqrt{2}$ (4) $2\pi(\sqrt{2}-1)$

If $\int_1^2 \frac{(x^2-1)dx}{x^3\sqrt{2x^4-2x^2+1}} = \frac{1}{k}$ then $k$ is____.

Let f and g be continuous functions on [-20, 20] such that f(-x) = f(x), g(-x) = -g(x), and g(x) = 1 + f(x)·f(-x). If $\int_{-20}^{20} f(x)\,dx = 2020$, then $I = \int_{-20}^{20} \dfrac{f(x)}{g(x)}\,dx$ equals:

Find $\displaystyle\lim_{n \to \infty} \left[\frac{1}{n^2}\sec^2\!\left(\frac{1}{n^2}\right) + \frac{2}{n^2}\sec^2\!\left(\frac{4}{n^2}\right) + \cdots + \frac{1}{n^2}\sec^2 1\right]$.

The value of the integral $\int_{-\pi/2}^{\pi/2} \sin^4 x \left(1 + \log\left(\dfrac{2+\sin x}{2-\sin x}\right)\right) dx$ is

Integrate: $\int_{-1.5}^{3.5} |x-1|\, dx$.

If the value of the definite integral $\int_0^1 {^{207}}C_x x^{200}(1-x)^7 dx$ is equal to $1/k$ where $k \in \mathbb{N}$, then the value of $k/26$ is ____.

If $f(x)+f(1-x)=1$ for all $x$, find $\displaystyle\int_0^1 f(x)\,dx$. [JEE Main 2016]

The value of $\lim \limits_{n \rightarrow \infty} \frac{1}{n} \sum \limits_{r=0}^{2 n-1} \frac{n^2}{n^2+4 r^2}$ is:

If $[\;]$ denotes the greatest integer function, then the integral $\displaystyle\int_0^{\pi}[\cos x]\,dx$ is equal to

Evaluate: \[\lim_{n \to \infty} \left[ \ln\left(\sqrt[n]{\frac{4}{n^2}}\right) + \ln\left(\sqrt[n]{\frac{16}{n^2}}\right) + \ln\left(\sqrt[n]{\frac{36}{n^2}}\right) + \cdots + \ln\left(\sqrt[n]{\frac{4n^2}{n^2}}\right) \right]\]

The value of definite integral $\displaystyle\int_{\frac{-1}{\sqrt{3}}}^{\frac{1}{\sqrt{3}}} \frac{\cos^{-1}\!\left(\dfrac{2x}{1+x^2}\right) + \tan^{-1}\!\left(\dfrac{2x}{1-x^2}\right)}{e^x + 1}\,dx$ is equal to:

Question 586. The value of 'm' is equal to:

If $\displaystyle\int_0^{\pi/3} \frac{\tan\theta}{\sqrt{2k\sec\theta}}\, d\theta = 1 - \dfrac{1}{\sqrt{2}}$, $(k > 0)$, then the value of $k$ is:

Find $\int_{0}^{2} (x^2 + 1) dx$ as the limit of a sum.

Let f(x) = $\int_0^x g(t)\,dt$, where g is a non-zero even function. If f(x + 5) = g(x), then $\int_0^x f(t)\,dt$ equals:

If $I = \int_0^1 (1-x^4)^7\, dx$, then the value of $\dfrac{29\int_0^1(1-x^4)^7\,dx}{10\int_0^1(1-x^4)^6\,dx}$ is:

The value of $2^{2010} \frac{\int_0^1 x^{1004}(1-x)^{1004} dx}{\int_0^1 x^{1004}\left(1-x^{2010}\right)^{1004} dx}$ is ____.

Find a real number $c$ and a positive number $L$ for which $\displaystyle\lim_{r \to \infty} \dfrac{r^c \int_0^{\pi/2} x^r \sin x\,dx}{\int_0^{\pi/2} x^r \cos x\,dx} = L$

If $I_1 = \int_{0}^{1} 2x^2 \, dx$, $I_2 = \int_{0}^{1} 2x^3 \, dx$, $I_3 = \int_{1}^{2} 2x^2 \, dx$ and $I_4 = \int_{1}^{2} 2x^3 \, dx$, then

Evaluate $\displaystyle\int_0^{\pi/2}\frac{\sin x-\cos x}{1+\sin x\cos x}\,dx$ [JEE Main 2016]

If the system of equations 2x - y + z = 0, x + 2y + z = 0, tx - y + 2z = 0 has infinitely many solutions and f(x) be a continuous function such that f(5 - x) + f(x) = 2, then \int_{0}^{2t} f(x)\,dx is equal to

If $m$ and $n$ are positive integers and $\int_a^b (t - a)^{2n} ((t - b)^{2m+1} \, dt$, \(a

$\displaystyle\lim_{x\to0}\dfrac{48}{x^4}\int_0^x\dfrac{t^3}{t^6+1}\,dt$ is equal to ___.

Given $ I = \int_a^b (x^4 - 2x^2)\,dx $ and $ f(x) = x^2(x^2 - 2) $. Find the ordered pair $(a, b)$ such that $I$ is minimum.

If $f(2-x) = f(2+x)$ and $f(4-x) = f(4+x)$ and $f(x)$ is a function for which $\int_0^2 f(x)\,dx = 5$, then $\int_0^{50} f(x)\,dx$ is equal to

The value of $\int_{-2}^{3} |1-x^2|\, dx$ is:

We have \[I = \int_0^\pi x f(\sin x)\,dx\] Using the property of definite integrals, find the value of $I$.

Evaluate $\int_0^{\pi/2} \dfrac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}}\, dx$

Let function $F$ be defined as $F(x) = \displaystyle\int_1^x \dfrac{e^t}{t}\,dt,\; x > 0$, then the value of the integral $\displaystyle\int_1^x \dfrac{e^t}{t+a}\,dt$, where $a > 0$, is

If $ I_n = \displaystyle\int_0^{\pi/2} \dfrac{\sin^2 nx}{\sin x} \, dx $, then $ I_{n+1} + I_{n-1} - 2I_n $ equals:

$\int_0^{\pi} [\cot x]\, dx$, where $[\cdot]$ denotes the greatest integer function, is equal to

Evaluate$\int_2^3 2 x^2 e^{x^3} d x$.

The integral $\displaystyle\int_0^{1/2}\dfrac{\ln(1+2x)}{1+4x^2}\,dx$ equals:

The value of $\int_0^{\pi} \sqrt{1 + 4\sin^2\dfrac{x}{2} - 4\sin\dfrac{x}{2}}\,dx$ is:

Evaluate $\int_0^1 \dfrac{\ln(x+1)}{x^2+1}\,dx$

Definite Integration Questions (1340)