Definite Integration Questions (1340)

If $a \leq \int_0^1 \frac{dx}{\sqrt{4-x^2-x^3}} \leq b$, then $(a,b) =$

$\displaystyle\lim_{n\to\infty}\dfrac{3}{n}\left\{4+\left(2+\dfrac{1}{n}\right)^2+\left(2+\dfrac{2}{n}\right)^2+\cdots+\left(3-\dfrac{1}{n}\right)^2\right\}$ is equal to:

Let $I = \int_{\pi/4}^{\pi/3} \frac{\sin x}{x} dx$, then $I$ belongs to:

Let $S = \left\{(x,y): \frac{y(3x-1)}{x(3x-2)} < 0\right\}$, $S' = \{(x,y): A \times B: -1 \leq A \leq 1, -1 \leq B \leq 1\}$, then the area of the region enclosed by all points in $S \cap S'$ is____.

Let $f(x)$ be a continuous function and 'c' is a constant satisfying $\int_0^x f(t) dt = e^x - ce^{2x} \int_0^x f(t)^2 dt$, then:

The value of $\int_0^{\pi/2} \log(\sin^2 \theta + k^2 \cos^2 \theta) d\theta$, where $k \geq 0$, is:

If $\Lim_{n \to \infty} \frac{1}{n^2} \sum_{k=1}^{n-1} k \left[ \int_0^{k/n} \sqrt{(x-k)(k+1-x)} dx \right] = \frac{\pi}{m^n}$, then:

Let $I(x)=\displaystyle\int\dfrac{x^2(x\sec^2x+\tan x)}{(x\tan x+1)^2}\,dx$. If $I(0)=0$, then $I\!\left(\dfrac{\pi}{4}\right)$ is equal to

The value of $\frac{dI}{da}$ when $I = \int_0^{\pi/2} \log\left(\frac{1 + a \sin x}{1 - a \sin x}\right) \frac{dx}{\sin x}$ (where $|a| < 1$) is:

If \(a>0\), evaluate \(\displaystyle\int_{-\pi}^{\pi}\frac{\cos^2 x}{1+a^x}\,dx\) [JEE Main 2020]

If $f(x)$ is an even function, then:

Let $f(n, z) = \int \cos(nz) dx$, with $f(0, 0) = 0$. If the expression $\sum_{i=1}^{n} f(i, 1)$ simplifies to $\frac{\sin(a+b)}{\sin c}$, then the value of $\frac{a}{c}$ is (where $a > b$)

Let $u = \int_0^{\pi/4} \left(\frac{\cos x}{\sin x + \cos x}\right)^2 dx$ and $v = \int_0^{\pi/4} \left(\frac{\sin x + \cos x}{\cos x}\right)^2 dx$, then:

For $\alpha,\beta,\gamma,\delta\in\mathbb{N}$, if $\displaystyle\int\!\left[\left(\frac{x}{e}\right)^{2x}+\left(\frac{e}{x}\right)^{2x}\right]\ln x\,dx=\dfrac{1}{\alpha}\!\left(\frac{x}{e}\right)^{\beta x}-\dfrac{1}{\gamma}\!\left(\frac{e}{x}\right)^{\delta x}+C$, then $\alpha+2\beta+3\gamma-4\delta$ is equal to

The sum of the series as $n \to \infty$ $\frac{\sqrt{n}}{(3+4\sqrt{n})^2} + \frac{\sqrt{n}}{\sqrt{2}(3\sqrt{2}+4\sqrt{n})^2} + \frac{\sqrt{n}}{\sqrt{3}(3\sqrt{3}+4\sqrt{n})^2} + \ldots + \frac{1}{49n}$ is:

The value of $\int \cot^{-1}\left(\frac{x^2+x+1}{1-x-x^2}\right)dx$ is equal to

If $I_n = \int \cot^n x dx$, then $I_0 + I_1 + 2(I_2 + I_3 + ... + I_k) + I_9 + I_{10} =$

The integral $\int \frac{\sec^{3/2}\theta - \sec^{1/2}\theta}{2 + \tan^2\theta} \tan\theta d\theta$ is:

The value of the integral $I = \int_0^1 t^{(1-t)} (1 + \ln t) dt$ is equal to

The integral $16\displaystyle\int_1^2\dfrac{dx}{x^3(x^2+2)^2}$ is equal to:

The value of $\lim_{n \to \infty} \sum_{r=1}^n \left(\frac{r}{n}\right)^2 \frac{1}{n}$ is equal to

If $\int_0^1 \frac{\sin t}{1+t} dt = a$, then the value of $\int_{4\pi-2}^{4\pi} \frac{\sin t}{4\pi + 2 - t} dt$ is:

The value of the integral $\displaystyle\int_{-\log_e 2}^{\log_e 2} e^x\!\left(\log_e(e^x+\sqrt{1+e^{2x}})\right)dx$ is equal to

If $G(x,t) = \begin{cases} x(t-1), & \text{when } x \leq t \\ t(x-1), & \text{when } t < x \end{cases}$ and if $f$ is continuous function of $x$ in $[0,1]$. Let $g(x) = \int_0^1 f(t)G(x,t)dt$. Then which is incorrect:

If $A = \int_0^{\sin \theta} \frac{t dt}{1 + t^2}$ and $B = \int_0^{\cos \theta} \frac{dt}{t(1 + t^2)}$, then the value of $e^A e^B \begin{vmatrix} A & A^2 & B \\ 1 & B^2 & -1 \\ 1 & A^2 + B^2 & -1 \end{vmatrix}$ is:

Evaluate \(\displaystyle\int_0^1 x^2 e^x\,dx\) [JEE Main 2018]

If \(f\) is continuous, prove/use: \(\displaystyle\int_0^{\pi} x\,f(\sin x)\,dx = \frac{\pi}{2}\int_0^{\pi} f(\sin x)\,dx\). Hence evaluate \(\displaystyle\int_0^{\pi}\frac{x\sin x}{1+\cos^2 x}\,dx\)

Evaluate \(\displaystyle\int_1^e(\ln x)^2\,dx\) [JEE Main 2019]

The value of the integral $\displaystyle\int_{-\pi/4}^{\pi/4}\dfrac{x+\pi/4}{2-\cos2x}\,dx$ is:

Let $I(x)=\displaystyle\int\frac{3\,dx}{(4z+6)\left(\sqrt{4x^2+8x+3}\right)}$ and $I(0)=\dfrac{\sqrt{3}}{4}+20$. If $I\!\left(\dfrac{1}{2}\right)=\dfrac{a\sqrt{2}}{b}+c$, where $a,b,c\in\mathbb{N}$, $\gcd(a,b)=1$, then $a+b+c$ is equal to

Evaluate \(\displaystyle\int_{-1}^2|x^2-x|\,dx\) [JEE Main 2014]

Consider $A = \int_0^{\pi/2} \frac{\sin(2x)}{1}dx$, then

$\displaystyle\int_{\frac{3\sqrt{2}}{4}}^{\frac{3\sqrt{3}}{4}}\dfrac{48}{\sqrt{9-4x^2}}\,dx$ is equal to:

Evaluate \(\displaystyle\int_0^{\pi/2}\frac{\sin^2 x}{\sin x+\cos x}\,dx\) [JEE Main 2022]

Consider the integrals $I_1 = \int_0^1 e^{-x}\cos^2 x dx, I_2 = \int_0^1 e^{-x^2} \cos^2 x dx, I_3 = \int_0^1 e^{-x} dx$ and $I_4 = \int_0^1 e^{-2/2} dx$. Then:

If $\int_x^y f(t) dt$ is independent of $x$ and $f(2) = 2$, if the value of $\int_2^x f(t) dt = k \ln x$ then $k$ is ____.

For $a \geq 2$, if the value of the definite integral $\int_0^a \frac{dx}{a^2+(x-(1:x))^2}$ equals to $\frac{\pi}{5050}$ then $\frac{a}{25}$ is____.

If $$\int \frac{dx}{x^2(x^n + 1)^{(n-1)/n}} = -[f(x)]^{1/n} + C$$, then $f(x)$ is

If \(f(x) = \begin{vmatrix} \sin x + \cos x & \ln(\cos x) & 1 + (\tan x)^2 \\ \pi & \pi^2 & \pi^4 \\ 1 & 7 & 1 \end{vmatrix}\), the value of \(\int_0^{\pi/2} f(x) dx\) is

Given that I(n) = \(\int \limits_{1}^{e}\)(1 + log x)n dx, n \(\in\) N satisfies I(n) = 2n e - 1 - nI(n - 1). The value of \(\int \limits_{1}^{e}\)(5 + log x)(1 + log x)2 dx is

Evaluate: \(\int \left( a^x \ln x + \ln a \cdot \ln\left(\frac{x}{e}\right) \right) dx\)

If \(\int \frac{\sin x}{\sin(x-a)} dx = Ax + B \log \sin(x-a) + C\), then the value of \((A, B)\) is:

Evaluate \(\displaystyle\int_0^1\tan^{-1}x\,dx\) [JEE Main 2017]

Evaluate the integral: \[\int_{1/3}^{1} \frac{\pi \cos\left(\dfrac{\pi}{2}x\right)}{2\sin^2\left(\dfrac{\pi}{2}x\right)}\, dx\]

The value of the definite integral \(\int_{1}^{7} \frac{\cos^2 x}{\cos^2 x + \cos^2(10-x)} dx\) is:

If \(\displaystyle\int\frac{x-1}{2x+1}\,dx = A\ln|2x+1|+Bx+C\), then

A sequence of non-negative terms is given by \(a_{n+1}=\int_\limits{\frac{2}{3}\left(a_{n-1}\right)}^{a_{n}} 3 d x\) where a0 = 0, a1 = 1. Then the value (a1 + a2 +a3 + ..... + a100) equals :

[JEE Main 2022] \(\displaystyle\int\frac{dx}{1+3\sin^2 x+8\cos^2 x}\) equals (where \(C\) is constant)

A differentiable function f is given by f(x) = \(\frac{1}{x^{2}} \int \limits_{4}^{x}\)(4t2 - 2f'(t))dt, x > 0. The value of f'(4) is

[JEE Main 2022] \(\displaystyle\int\frac{\sqrt{\tan x}+\sqrt{\cot x}}{\sqrt{\sin x}\cdot\sqrt{\cos x}}\,dx\) equals (where \(C\) is a constant)