List of top Mathematics Questions on Integration asked in JEE Main

The value of \[ \sum_{r=1}^{20}\sqrt{\left|\pi\left(\int_0^r x|\sin \pi x|\,dx\right)\right|} \] is:

If \[ \int (\cos x)^{-5/2}(\sin x)^{-11/2}\,dx = \frac{p_1}{q_1}(\cot x)^{9/2} + \frac{p_2}{q_2}(\cot x)^{5/2} + \frac{p_3}{q_3}(\cot x)^{1/2} - \frac{p_4}{q_4}(\cot x)^{-3/2} + C, \] where $C$ is the constant of integration, then find the value of \[ \frac{15\,p_1p_2p_3p_4}{q_1q_2q_3q_4}. \]

If a line x = -1 divides the area of region bounded by $\{(x,y):1+x^2 \leq y \leq 3-x\}$ in the ratio $\frac{m}{n}$ then (m + n) equal (where HCF of (m,n) = 1) :

If \[ \int \frac{1-5\cos^2 x}{\sin^5 x\cos^2 x}\,dx=f(x)+C, \] where $ C $ is the constant of integration, then \[ f\!\left(\frac{\pi}{6}\right)-f\!\left(\frac{\pi}{4}\right) \] is equal to:

The value of
\[ \left(\frac{1}{3}+\frac{4}{7}\right) +\left(\frac{1}{3^2}+\frac{1}{3}\times\frac{4}{7}+\frac{4}{7^2}\right) +\left(\frac{1}{3^3}+\frac{1}{3^2}\times\frac{4}{7}+\frac{1}{3}\times\frac{4}{7^2}+\frac{4}{7^3}\right) +\cdots \text{ up to infinite terms is} \]

The sum of all possible values of $ n \in \mathbb{N} $, so that the coefficients of $x, x^2$ and $x^3$ in the expansion of $(1+x^2)^2(1+x)^n$ are in arithmetic progression is :

Let \[ f(t)=\int \left(\frac{1-\sin(\log_e t)}{1-\cos(\log_e t)}\right)dt,\; t>1. \] If $f(e^{\pi/2})=-e^{\pi/2}$ and $f(e^{\pi/4})=\alpha e^{\pi/4}$, then $\alpha$ equals

Let a differentiable function $f$ satisfy \[ \int_0^{36} f\!\left(\frac{tx}{36}\right)dt=4\alpha f(x). \] If $y=f(x)$ is a standard parabola passing through the points $(2,1)$ and $(-4,\beta)$, then $\beta^2$ is equal to

Let $(2\alpha,\alpha)$ be the largest interval in which the function \[ f(t)=\frac{|t+1|}{t^2},\; t<0 \] is strictly decreasing. Then the local maximum value of the function \[ g(x)=2\log_e(x-2)+\alpha x^2+4x-\alpha,\; x>2 \] is

From the first 100 natural numbers, two numbers first $ a $ and then $ b $ are selected randomly without replacement. If the probability that $ a - b \ge 10 $ is $ m/n $, $ \text{gcd}(m, n) = 1 $, then $ m + n $ is equal to :

Let $[\,]$ be the greatest integer function. If \[ \alpha=\int_{0}^{64}\left(x^{1/3}-[x^{1/3}]\right)\,dx, \] then \[ \frac{1}{\pi}\int_{0}^{\alpha\pi}\frac{\sin^2\theta}{\sin^6\theta+\cos^6\theta}\,d\theta \] is equal to ____________.

If the area of the region \[ \{(x,y): x^2+1 \le y \le 3-x\} \] is divided by the line $x=-1$ in the ratio $m:n$ (where $m$ and $n$ are coprime natural numbers), then find the value of $m+n$.

Area enclosed by $ 4x^2 + y^2 \le 8 $ and $ y^2 \le 4x $ (in square unit) is:

The value of $ \int_{-\pi/2}^{\pi/2} \frac{dx}{[x]+4} $ is, where [.] denotes greatest integer function:

If $\int(\sin x)^{-11/2} (\cos x)^{-5/2} dx = \frac{p_1}{q_1}(\cot x)^{9/2} - \frac{p_2}{q_2}(\cot x)^{5/2} - \frac{p_3}{q_3}(\cot x)^{1/2} + \frac{p_4}{q_4}(\cot x)^{-3/2} + C$ where H.C.F. $\{p_i, q_i\} = 1$ \& $i \in \{1, 2, 3, 4\}$. Then value of $\sum_{i=1}^{4}(p_i +q_i)$ is:

Let $ A = [a_{ij}] $ be a square matrix of order 2 with entries either 0 or 1. Let $ E $ be the event that $ A $ is an invertible matrix. Then the probability $ P(E) $ is:

The integral $ \int_0^\pi \frac{(x + 3) \sin x}{1 + 3 \cos^2 x} \, dx $ is equal to:

The value of \[ \int_{e^2}^{e^4} \frac{1}{x} \left( \frac{e^{\left( (\log_e x)^2 +1 \right)^{-1}}}{e^{\left( (\log_e x)^2 +1 \right)^{-1}} + e^{\left( (6-\log_e x)^2 +1 \right)^{-1}}} \right) dx \] is:

Let (a, b) be the point of intersection of the curve $x^2 = 2y$ and the straight line $y - 2x - 6 = 0$ in the second quadrant. Then the integral $I = \int_{a}^{b} \frac{9x^2}{1+5^{x}} \, dx$ is equal to:

If the function \[ f(x) = \begin{cases} \frac{2}{x} \left( \sin(k_1 + 1)x + \sin(k_2 -1)x \right), & x<0 \\ 4, & x = 0 \\ \frac{2}{x} \log_e \left( \frac{2 + k_1 x}{2 + k_2 x} \right), & x>0 \end{cases} \] is continuous at $ x = 0 $, then $ k_1^2 + k_2^2 $ is equal to:

If \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{96x^2 \cos^2 x}{1 + e^x} dx = \pi(a\pi^2 + \beta), \quad a, \beta \in \mathbb{Z}, \] then $ (a + \beta)^2 $ equals:

Let \[ I(x) = \int \frac{dx}{(x-11)^{\frac{11}{13}} (x+15)^{\frac{15}{13}}} \] If \[ I(37) - I(24) = \frac{1}{4} \left( b^{\frac{1}{13}} - c^{\frac{1}{13}} \right) \] where $ b, c \in \mathbb{N} $, then \[ 3(b + c) \] is equal to:

$$ \int \frac{1}{\sqrt{3+x^2}+\sqrt{1+x^2}} \, dx - 3 \log \left( \sqrt{3} \right) $$ is equal to:

If $ \int \left( \frac{1}{x} + \frac{1}{x^3} \right) \left( \sqrt[23]{3x^{-24}} + x^{-26} \right) \, dx $ is equal to $ -\frac{\alpha}{3(\alpha + 1)} \left( 3x^\beta + x^\gamma \right)^{\alpha + 1} + C, \quad x>0, $ where $ \alpha, \beta, \gamma \in \mathbb{Z} $ and $ C $ is the constant of integration, then $ \alpha + \beta + \gamma $ is equal to _______.

Let $ f(x) = \int x^3 \sqrt{3-x^2} dx $. If $ 5f(\sqrt{2}) = -4 $, then $ f(1) $ is equal to