Definite Integration Questions - JEE

If $\displaystyle\int_0^1 f(x)\,dx = 1 + 2\int_0^1 x\,f(x)\,dx$, and $f(x)=1+cx$, find $c$. [JEE Main 2021]

If ∫ (x-1) dxx22x2-2x+1 = f(x)g(x) + C, where f(x) is a quadratic expression and g(x) is a monic linear expression.

If $\int \frac{(x-1) dx}{x^2 \sqrt{2x^2 - 2x + 1}} = \frac{\sqrt{f(x)}}{g(x)} + C$, where $f(x)$ is a quadratic expression and $g(x)$ is a monic linear expression.

If $f\left(\frac{1-x}{1+x}\right)=x$ and $g(x)=\int f(x)dx$ then(A) $g(x)$ is continuous in domain(B) $g(x)$ is discontinuous at two points in its domain(C) $\lim_{x\to\infty} g'(x)=-1$(D) $\int g(x)dx=-\frac{x^2}{2}+(2x+1)\ln\left(\frac{1+x}{e}\right)+C$

The value of $\sum_{n=1}^{100} \int_{n-1}^{n} e^{x-[x]} dx$, where $[x]$ is the greatest integer $\leq x$, is (JEE Main 2021)

Let $A_n = \int \tan^n x \, dx$, $\forall n \in \mathbb{N}$. If $A_{n} + A_{n-2} = \frac{\tan^{n-1} x}{n-1} + \lambda$ (where $\lambda$ is an arbitrary constant), then the value of $m$ is equal to

The value of the integral $\int e^{x+1}(2x^2 - \frac{1}{x} + 1) dx$ is equal to (where $C$ is the constant of integration)

If $\int \frac{2\sin x}{\ln(1 + (f(x)) + C}$ (where $x > 0$ and $C$ is the constant of integration) then the range of $f(x)$ is

$\int e^{x\left(1 + n \cdot x^{n-1} + x^{2n}\right)} \frac{1}{(1+x^n)\sqrt{1+x^{2n}}} dx$ is equal to

The value of $\displaystyle\int_{-\pi/2}^{\pi/2}\frac{1}{[|x|]+4}\,dx$, where $[\cdot]$ denotes the greatest integer function, is

Evaluate $\lim_{n \to \infty} \left[ \frac{\sqrt{n}}{(3+4\sqrt{n})^2} + \frac{\sqrt{n}}{\sqrt{2}(3\sqrt{2}+4\sqrt{n})^2} + \frac{\sqrt{n}}{\sqrt{3}(3\sqrt{3}+4\sqrt{n})^2} + \dots + \frac{1}{49n} \right]$

Evaluate $\int \frac{dx}{\sqrt{4 - x^2}}$

The value of the integral $\int \frac{(x)^5}{x^n(1+x^n)^{1/n}} dx$, $n \in \mathbb{N}$ is

Let the value of integral is $I$, then, $I = \int_0^\pi (\cos 2x \cos 2^2 x \cos 2^3 x)dx$

The value of $\int_{\pi/4}^{\pi/4} \frac{d}{dx}\left(\frac{\cot x}{\cot x + \tan x}\right) dx$ is

Let I(x) = ∫ 6 / (sin^2 x (1 - cot x)^2) dx. If I(0) = 3, then I(π/12) is equal to :

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 $\int_{1}^{2} (x^{[x^2]} + [x^2]^x) dx$, where $[.]$ denotes the greatest integer function, is equal to -

Let $p(x)$ be a function defined on $\mathbb{R}$ such that $p'(x) = p'(1-x)$, for all $x \in [0,1]$, $p(0) = 1$ and $p(1) = 41$. Then $\int_0^1 p(x)\, dx$ equals

The value of the integral $\int_1^e \frac{\log_e x}{x} dx$ is

If $f(x) = \begin{cases} e^{\sin x} & \text{for } |x| \leq \pi/2 \\ 2 & \text{otherwise} \end{cases}$, then $\int_{-\pi}^{\pi} \frac{2f(x)}{\cos x} dx$ is

Integral of $\sqrt{1+2\cot x(\cot x+\csc x)}$ w.r.t. $x$ is

The integral ∫(sin(101x)·sin^99 x)dx equals

If I_n = ∫(sin x)^n dx n ∈ N, then 5I_4 - 6I_6 is equal to -

If $F(x) = \displaystyle\int_x^{x^2}(t^2+1)\,dt$, find $F'(x)$.

Let $ f(x) $ be a continuous function $ \forall\, x \in \mathbb{R} $ such that \[ \lim_{x \to \pi/4} \frac{\displaystyle\int_{2}^{\sec^2 x} f(t)\, dt}{x^2 - \dfrac{\pi^2}{16}} = \frac{k}{\pi} f(a) \] where $ a, k \in \mathbb{N} $, then the value of $ k^a $ is equal to:

If $m = \displaystyle\int_{-2}^{0} \frac{|\sin x|}{\left[\dfrac{x}{\pi}\right] + \dfrac{1}{2}}\,dx$ and $n = \displaystyle\int_0^2 \frac{|\sin x|}{\left[\dfrac{x}{\pi}\right] + \dfrac{1}{2}}\,dx$, where [ ] is the greatest integer function, then $-\dfrac{m}{n}$ = ______.

Evaluate the following: 21. $\int_0^{1/n} \tan^{-1}\sqrt{Vx-1} \, dx$

Let $f:(0,\infty)\to R$ and $F(x)=\displaystyle\int_0^x t\,f(t)\,dt$. If $F(x^2)=x^4+x^5$, then $\displaystyle\sum_{r=1}^{12} f(r^2)$ is equal to $k$. Find $\dfrac{k}{73}$.

If $f(x) + g(x) + h(x) = 2 \ \forall \ x \in R$, then the value of the expression $\displaystyle\int_0^{3/4} \left(f^2(x) + g^2(x) + h^2(x)\right) dx$, can be:

Evaluate: $\displaystyle\int_{-\pi/2}^{\pi/2} \frac{2}{1+e^x}\,dx$ = ______ (up to four decimal places).

Integrate $\int \dfrac{dx}{(3+x^2)\sqrt{1-x}}$.

Consider $I(a) = \int_0^a \frac{dx}{x}$ (where $a > 0$), then the value of $\sum_{i=1}^n I(i) + \sum_{i=1}^{n-1} I(i)$ is

The value of $\int_0^{\pi/3} \log(1 + \sqrt{3}\tan x)dx$ is equal to

Evaluate the following: 30. $\int_0^{2\pi} \sqrt{2ax - x^2} \, dx$

Evaluate $\displaystyle\int_{-1}^{3} |\sin \pi x|\, dx$.

Evaluate $\int \sqrt{1 + \sec x}\, dx$

If $f(x) + g(x) + h(x) = 2$ $\forall\, x \in R$, then the value of the expression $\displaystyle\int_0^{3/4} \left(f^2(x) + g^2(x) + h^2(x)\right) dx$ can be:

If $\displaystyle\int_0^{\pi} \frac{\sin x(\sin x + 1)e^{\sin x + \cos x}}{e^{\cos x} + 1}\, dx = a + b\int_0^{\pi} e^{\sin x}\, dx$ where $a$ and $b$ are positive rational numbers, then find the value of $100(a^2 + b^2)$.

Evaluate: \ 11\int \frac{\sec^2 x}{(\sec x + \tan x)^n} \, dx, \quad (n > 1)

Evaluate: \ 3\int \frac{\sin^{-1}\sqrt{x}}{\sqrt{1-x}} \, dx

Evaluate $\displaystyle\int_{-1}^1 xe^{|x|}\,dx$ [JEE Main 2022]

If $I_1 = \int_0^1 x\sin(1-x)dx$ and $I_2 = \int_0^1 x\sin(1-x)dx$, then $\frac{I_2}{I_1}$ is equal to

The value of $\int_{-\pi}^{\pi} [\sin(1-x)dx]$ is equal to (where $[.]$ denotes the greatest integer function)

Let $I_1 = \int_0^1 \frac{x^2}{x^4+1}dx$ and $I_2 = \int_0^1 \frac{\log(x+\frac{1}{x})}{1+x^2}dx$, then

Suppose f and g are differentiable functions such that $xg(f(x))f'(g(x))g'(x) = f(g(x))g'(f(x))f'(x)$ for all real $x$. Also $f$ is non negative and $g$ is positive. If $\int_0^a f(g(x))\,dx = \frac{1}{2} - \frac{e^{-2a}}{2}$ for all reals $a$ and $g(f(0)) = 1$, then the value of $g(f(4))$ is equal to $e^{-\lambda}$ where $\lambda \in N$. Find the value of $\lambda$.

549. If the system of equation $2x - y + z = 0$, $x - 2y + z = 0$ 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:

Evaluate $\displaystyle\int_0^{\pi/4}\sec^3 x\,dx$ [JEE Main 2019]

Evaluate $\displaystyle\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n\frac{1}{1+k/n}$ [JEE Main 2016]

Definite Integration Questions (1340)