Definite Integration Questions - JEE

If the integral ∫ 5tanx dx/tanx-2 = x + aln|sinx - 2cosx| + C, then a is equal to:

70. $\int \frac{x^2}{x^2 + 1} \cdot \frac{x - 1}{x + 1} \, dx$ equals

Let $I_1 = \int_0^{10^4} \frac{\{\sqrt{x}\}}{\sqrt{x}}\,dx$ and $I_2 = \int_0^{10} \left(x\{x^2\}\right)dx$, where $\{.\}$ denotes fractional part of $x$. Then

Let $f: {R} \rightarrow {R}$ be a twice differentiable function such that $f(2)=1$. If ${F}(x)=x f(x)$ for all, $x \in {R}, \int_{0}^{2} x F^{\prime}(x) d x=6$ and $\int_{0}^{2} x^{2} F^{\prime \prime}(x) d x=40$, then $F^{\prime}(2)+\int_{0}^{2} F(x) d x$ is equal to:

The value of $\int_0^2 (px^3 + qx + r)\, dx$, where $p, q, r$ are constants, depends on the value of

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

The value of $\int_0^1 \frac{\sin^{-1} x}{\sqrt{1-x^2}} dx$ is

256. $\displaystyle\int_0^{10} [x]^3 \{x\}\, dx$ is equal to:[Note: Where $[\,]$ and $\{\,\}$ denotes greatest integer and fractional part functions respectively]

The value of the definite integral $\displaystyle\int_0^{\pi/4} \dfrac{\sin^3 x \cos^3 x}{(\sin^4 x + \cos^4 x)^2}\,dx$ is equal to:

Evaluate $\displaystyle\int_0^\infty x\,e^{-x^2}\,dx$ [JEE Main 2020]

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]

Definite Integration Questions (1340)