Definite Integration Questions (1340)

Evaluate \(\displaystyle\int_0^1\frac{x}{(x^2+1)^2}\,dx\) [JEE Main 2020]

Evaluate \(\displaystyle\lim_{n\to\infty}\frac{1^p+2^p+\cdots+n^p}{n^{p+1}}\), \(p>0\) [JEE Main 2014]

Evaluate \(\displaystyle\int_{-\pi}^{\pi}\frac{x^3 + x\cos x + \tan^5 x}{2+\cos^2 x}\,dx\)

Let \( I = \int_0^{\pi/2} \log \tan x \, dx \). Find the value of \(I\).

Evaluate \(\displaystyle\int_0^{\pi}x\sin^3 x\,dx\)

Evaluate \(\displaystyle\int_0^2\frac{x^2+1}{x^4-x^2+1}\,dx\) [JEE Main 2022]

Let \(f: R \to R\) and \(g: R \to R\) be continuous functions. Then the value of \(\int_{-\pi/2}^{\pi/2} [f(x) + f(-x)][g(x) - g(-x)] dx\) is

Let $f$ be a differentiable function defined on $\left[0,\dfrac{\pi}{2}\right]$ such that $f(x)>0$ and $f(x)+\displaystyle\int_0^x f(t)\sqrt{1-(\log_e f(t))^2}\,dt=e$, $\forall x\in\left[0,\dfrac{\pi}{2}\right]$. Then $\left(6\log_e f\!\left(\dfrac{\pi}{6}\right)\right)^2$ is equal to ___.

The value of definite integral \(\displaystyle\int_{-\pi/4}^{\pi/4} \dfrac{x^2(f(x)+3)+1}{2g^2(x)+1}\, dx\) is:

Find $\int \frac{x^3}{\sqrt{x^2+2}} dx$

If \(f(x)\) is a continuous function for all real values of \(x\) and satisfies \(\int_n^{n+1} f(x) dx = 2\), \(\forall n \in \mathbb{I}\), then \(-\int_3^5 f(|x|) dx\) is equal to

For a positive integer \(n\), let \(I_n = \int_{-\pi}^{\pi} \left(\frac{\pi}{2} - |x|\right) \cos(nx) dx\). Find the value of \([I_1 + I_2 + I_3 + I_4]\) where \([\cdot]\) denotes the greatest integer function.

Maximum value of the function f(x) = π²∫₀¹ t sin(x + πt) dt over all real numbers x:

Let \(f(x) = \lim_{n \to \infty} \dfrac{\cos x}{1 + (\tan^{-1} x)^n}\), then the value of \(\displaystyle\int_0^{\infty} f(x)\, dx\) is equal to:

Evaluate \(\displaystyle\int_0^1|2x-1|\,dx\) [JEE Main 2015]

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:

The value of \(\lim_{x \to 0} \dfrac{\int_0^{x^2} \sec^2 t\, dt}{x \sin x}\) is

Evaluate $\int \ln(2x+3)^{(2x+3)} \, dx$

If $[t]$ denotes the greatest integer $\leq t$, then the value of $\dfrac{3(e-1)^2}{e}\displaystyle\int_1^2 x^2e^{[x]+[x^3]}\,dx$ is:

34. If the value of \(\lim_{n \to \infty} \sum_{k=0}^{n} \dfrac{{}^{n}C_k}{n^k(k+3)}\) equals \(L\). Then \([L]\) is equal to:[Note: Where \([k]\) denotes greatest integer function less than or equal to \(k\).]

Evaluate \(\displaystyle\int_0^1\sin\\!\left(2\tan^{-1}\\!\sqrt{\frac{1-x}{1+x}}\right)dx\) [JEE Main 2021]

If \(f(x) = a\cos(\pi x) + b\), \(f'\left(\frac{1}{2}\right) = \frac{\pi\sqrt{3}}{2}\) and \(\int_{1/2}^{3/2} f(x)\,dx = 2 + \frac{1}{\pi}\), then find the value of \(-\frac{12}{\pi}\left(\frac{\sin^{-1}a}{\sqrt{3}} + \cos^{-1}b\right)\).

The value of $12\displaystyle\int_0^3|x^2-3x+2|\,dx$ is ___.

Let \(f\) be a continuous and even function such that \(\int_0^a f(x)\,dx = 10\). If \(g(x)\) is a continuous positive function such that \(g(x)g(-x) = 1\) and \(\int_0^a g(x)\,dx = 5\), then find the value of \(\int_{-a}^{a} \dfrac{f(x)}{1+g(x)}\,dx\).

Evaluate \(\displaystyle\int_{-\pi/4}^{\pi/4} \frac{x + \frac{\pi}{4}}{2 - \cos 2x}\, dx\).

Evaluate \(\displaystyle\int_0^{2\pi} [\cos x]\,dx\) where \([\cdot]\) denotes the greatest integer function.

If \(I = 98\int_0^1 (\sin 2)e^x(\tan x + \sec^2 x)\,dx\) and \(I = p \cdot e \cdot \sin^2 q\), find \(p + q\).

Evaluate $\int_{0}^{4\pi} |\cos x| dx$

Evaluate \(\displaystyle\int_0^{10\pi}|\sin x|\,dx\) [JEE Main 2020]

Let $f(x)=[x]^2-[x+3]-3$, $x\in\mathbf{R}$, where $[\cdot]$ is the greatest integer function. Then

\(\displaystyle\int_0^{\pi/4}\frac{\sec^2 x}{(1+\tan x)^2}\,dx\) [JEE Main 2019]

Let $\int_\alpha^{\log_e4}\dfrac{dx}{\sqrt{e^x-1}}=\dfrac{\pi}{6}$. Then $e^\alpha$ and $e^{-\alpha}$ are the roots of the equation:

762. If \(\displaystyle\int_{x_1}^{x_2} \frac{f(x)f'(x)}{\sqrt{1-(f(x))^4}}\,dx \geq \int_{x_1}^{x_2} x\,dx\), where \(f(x_2^-) = \frac{1}{\sqrt{2}}\) and \(f(x_1^+) = 1\), then the minimum value of \(x_1^2 - x_2^2\) is \(\dfrac{k\pi}{3}\). Find \(k\).

If $f(x) = \begin{vmatrix} \cos x & e^{x^2} & 2x \cos^2 x/2 \\ x^2 & \sec x & \sin x + x^3 \\ 1 & 2 & x + \tan x \end{vmatrix}$, then the value of $\int_{-\pi/2}^{\pi/2} (x^2 + 1)\{f(x) + f''(x)\}dx$ is

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

Evaluate \(\displaystyle\int_0^1 xe^{-x}\,dx\) [JEE Main 2018]

Let $f(x)=\begin{cases}-2, & -2\leq x\leq0\\ x-2, & 0<x\leq2\end{cases}$ and $h(x)=f(|x|)+|f(x)|$. Then $\int_{-2}^{2}h(x)\,dx$ is equal to:

The minimum value of the function $f(x)=\displaystyle\int_0^2 e^{|x-t|}\,dt$ is:

\int_{-1}^{1} [x \sin(\pi x)]\,dx is equal to

Evaluate \(\displaystyle\int_0^{\pi/2}\frac{dx}{2+\cos x}\) [JEE Main 2018]

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 ____.