Definite Integration Questions - JEE

If ∫√(sec 2x - 1) dx = α log_e |cos 2x + β + √(cos 2x (1 + cos 2x / β))| + constant, then β - α is equal to ____.

For $ x \in (0,0) $, let $ I_3 = \int_0^1 e^{-x^3} dx $, $ I_2 = \int_0^1 e^{-x^2} \cos^2 x \, dx $, $ I_1 = \int_0^1 e^{-x} \cos^2 x \, dx $. Then which of the following is true?(1) $I_3 > I_2 > I_1$ (2) $I_1 > I_2 > I_3$ (3) $I_3 > I_2 > I_1$ (4) $I_2 > I_1 > I_3$

24. If $I = \displaystyle\int_{e^{\pi/6}}^{e^{\pi/2}} \dfrac{\sin(\ln(\sin(\ln x))) \cos(\ln x)}{x \sin(\ln x)}\, dx$, then the value of $\cos^{-1}(I + 1)$ is equal to:

If \[\int_0^x \prod_{r=1}^{2013} (1+r^2) dx = \left[\prod_{r=1}^{2013} (1+r^2) - k^2\right]\] then $k = $

Evaluate $\displaystyle\int_0^3[x^2]\,dx$ where $[\cdot]$ is GIF. [JEE Main 2017]

If $\int_0^x \cos t^2\, dt = \int_0^x \frac{\sin t}{t}\, dt$, find $\frac{dy}{dx}$ w.r.t. $z$.

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

The value of the integral $\int_0^2 \frac{\log(x^2+2)}{(x^2+2)} dx$ is:

Let $ I = \int_{-\pi/2}^{\pi/2} \sin^4 x \left(1 + \log\left(\dfrac{2+\sin x}{2-\sin x}\right)\right) dx $. Find the value of $I$.

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

Let g(x) be an antiderivative for f(x). Then ln(1+(g(x))^2) is an antiderivative for

Let $f: R \to R$ be continuous function and $f(x) = f(2x)$ is true $\forall\, x \in R$ and $f(1) = 3$, then the value of $\displaystyle\int_{-1}^{1} f(f(x))\, dx$ is equal to:

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

The integral ∫√cot x e^√sin x √cos x dx equals

$\int \frac{x^2 + x}{(e^x + x + 1)^2} dx$ equals

Evaluate $\displaystyle\int_0^{n\pi}|\sin x|\,dx$ where $n\in\mathbb{N}$.

If $I_n = \int_1^e (\ln x)^n dx$, then $I_n + nI_{n-1}$ is equal to

The $L$ denotes the value of the definite integral $\displaystyle\int_0^1 \dfrac{1}{1+x^8}\,dx$, then which one of the following must be true?

Question 585. The value of M is:

The value of $\int_{1}^{2} \frac{x}{1+\lfloor x \rfloor} dx$ (where $[x]$ is greatest integer function) is

Let $L = \lim_{n \to \infty} \dfrac{1}{n^3} \sum_{k=1}^{n} k^2 e^{\frac{k}{n}}$, then find the value of $e - L$.

If $I = \int_0^{1/2} \dfrac{1}{\sqrt{1-x^{2n}}}\,dx$, for $n \ge 1$, then

If $C \in \frac{2}{3}$ and $J < 2$

If $f(a+b+1-x) = f(x)$ for all $x$, where $a$ and $b$ are fixed positive real numbers, then $\int_{a}^{b} x(f(x) + f(x+1)) dx$ is equal to

Let g be a differentiable function satisfying $\int_\limits{0}^{x}$ (x - t + 1) g(t) dt = x4 + x2 for all $x \geq 0$. The value of $\int_\limits{0}^{1} \frac{12}{g^{\prime}(x)+g(x)+10}$ dx is equal to :

For $c , find the value of \(c$ such that $\int_c^1 (8x^2 - x^5) \, dx = \frac{16}{3}$.

∫ sec2θ(secθ + tanθ)2dθ

The value of $\int_{3/4}^{2} |1 - x^2| dx$ is

Find the value of the definite integral \[\left(\frac{1}{\pi}\int_0^{\pi/2} \frac{\cos^4 x + \sin x\cos^3 x + \sin^2 x\cos^2 x + \sin^3 x\cos x}{\sin^4 x + \cos^4 x + 2\sin x\cos^3 x + 2\sin^2 x\cos^2 x + 2\sin^3 x\cos x}\,dx\right)^{-1}\]

Given $f(x) = \begin{cases} x[x] & \text{for } x \leq -1 \\ [x+1] + [1-x] & \text{for } -1 < x < 1 \\ x[x] & \text{for } x \geq 1 \end{cases}$ where $[\cdot]$ denotes the greatest integer function. If $I = \int_{-1}^{2} f(x) dx$, then $|3I| =$

Evaluate $\int_{0}^{3\pi/4} [(1+x)\sin x + (1-x)\cos x] dx$

Evaluate $\displaystyle\int_0^{\pi/2}\log(\tan x)\,dx$

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

Evaluate $\displaystyle\int_0^{\pi/2} \sin x \cdot \sin 2x \cdot \sin 3x\, dx$

Let $I(x)=\displaystyle\int\frac{6}{\sin^2 x(1-\cot x)^2}\,dx$. If $I(0)=3$, then $I\!\left(\dfrac{\pi}{12}\right)$ is equal to

Evaluate $\int_0^1 (tx + 1 - x)^n\, dx$, $n \in \mathbb{N}$ and $t$ is a parameter independent of $x$. Also show that \[\int_0^1 x^k (1-x)^{n-k}\, dx = \frac{1}{(n+1)\cdot {}^nC_k}.\]

Given $f: \mathbb{R} \to \mathbb{R}$ such that $f(2-x) = f(2+x)$ and $f(4-x) = f(4+x)$ for all $x \in \mathbb{R}$. If $\int_0^2 f(x)\,dx = 5$, find $\int_{10}^{50} f(x)\,dx$.

Let $f$ be integrable. Let $I_1 = \int_0^{\pi/2} \cos^2 x \cdot f(\sin x + \cos^2 x) dx$ and $I_2 = \int_0^{\pi/2} \sin^2 x \cdot f(\sin x + \cos^2 x) dx$, then

Let $I_1 = \int_0^1 x \times x^{49}(1-x^{50})^{100} dx$. If $I_2 = aI_1$, find the value of $a$.

If $\int \frac{1}{a^2 \sin^2 x + b^2 \cos^2 x} dx = \frac{1}{12} \tan^{-1}(3 \tan x) + \text{constant}$, then the maximum value of $a \sin x + b \cos x$ is :

If $\int_0^a f(x)dx = \int_f(2a-x) = -f(x)$, then $\int_0^a f(x)dx$ equals:

If $f(x) = \begin{cases} x+3 & : x < 3 \\ 3x^2+1 & : x \ge 3 \end{cases}$, then find $\int_2^5 f(x) dx$.

Given $\int_a^b f(x) \, dx = \int_b^a f(x) \, dx$ and $f'(x) \geq 0$ at any $x \in (a,b)$, with $f(x)$ being continuous and differentiable in $(a,b)$. If $g(x) = \int_0^x f(t) \, dt$, then $\int_a^b f(x)g(x) \, dx \geq 0$ implies

Evaluate $\displaystyle\int_0^1\frac{e^x}{1+e^x}\,dx$ [JEE Main 2019]

The value of $\int_0^{\pi/2} \ln(\sin^2\theta + k^2\cos^2\theta) d\theta$ is equal to

Let $f(x)=\left\{\begin{array}{cc}-2, & -2 \leq x \leq 0 \\ x-2, & 0 \lt x \leq 2\end{array}\right.$ and $h(x)=f(|x|)+|f(x)|$. Then $\int_{-2}^{2} h(x) d x$ is equal to:

If $\lim_{n \to \infty}\left(\dfrac{1}{\sqrt{n}\sqrt{n+1}} + \dfrac{1}{\sqrt{n}\sqrt{n+2}} + \cdots + \dfrac{1}{\sqrt{n}\sqrt{n+n}}\right)$ is equal to $a\sqrt{b} - \dfrac{c}{d}$, where a, b, c and d are positive integers, c and d are co-prime. Find the value of $(a^4 + b^3 + c^2 + d)$.

Evaluate $\displaystyle\int_1^4\frac{dx}{x+\sqrt{x}}$ [JEE Main 2019]

If $I = \int_0^1 \cos\!\left(\frac{\pi}{2}x\right)dx \cdot \int_0^1 \cos^2\!\left(\frac{\pi}{2}x\right)dx \cdot \int_0^1 \cos^3\!\left(\frac{\pi}{2}x\right)dx \cdot \int_0^1 \cos^4\!\left(\frac{\pi}{2}x\right)dx = \frac{k}{\pi^2}$, find the value of $\dfrac{1}{k}$.

Evaluate: \[\int_{-1/\sqrt{3}}^{1/\sqrt{3}} \frac{\cos^{-1}\!\left(\dfrac{2x}{1+x^2}\right)+\tan^{-1}\!\left(\dfrac{2x}{1-x^2}\right)}{e^x+1}\, dx\]

Definite Integration Questions (1340)