List of top Mathematics Questions on Integration

Let a line $L_1$ pass through the origin and be perpendicular to the lines $L_2: \vec{r} = (3 + t)\hat{i} + (2t - 1)\hat{j} + (2t + 4)\hat{k}$ and $L_3: \vec{r} = (3 + 2s)\hat{i} + (3 + 2s)\hat{j} + (2 + s)\hat{k}$. If $(a, b, c)$, $a \in \mathbb{Z}$, is the point on $L_3$ at a distance of $\sqrt{17}$ from the point of intersection of $L_1$ and $L_2$, then $(a + b + c)^2$ is equal to ________.

The value of the integral $\int_{\pi/6}^{\pi/3} \left( \frac{4 - \csc^2 x}{\cos^4 x} \right) dx$ is:

Let $ f(x) = \int_{} \frac{16x + 24}{x^2 + 2x - 15} \, dx $. If $ f(4) = 14 \log_e(3) $ and $ f(7) = \log_e(2^\alpha \cdot 3^\beta) $, where $ \alpha, \beta \in \mathbb{N} $, then $ \alpha + \beta $ is:

$ \int_{-\pi/4}^{\pi/4} \frac{32 \cos^4 \theta}{1 + e^{\sin \theta}} \, d\theta $ is equal to:

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 :

A racing track is built around an elliptical ground whose equation is given by \[ 9x^2 + 16y^2 = 144 \] The width of the track is $3$ m as shown. Based on the given information answer the following:

(i) Express $y$ as a function of $x$ from the given equation of ellipse.
(ii) Integrate the function obtained in (i) with respect to $x$.
(iii)(a) Find the area of the region enclosed within the elliptical ground excluding the track using integration.
OR
(iii)(b) Write the coordinates of the points $P$ and $Q$ where the outer edge of the track cuts $x$-axis and $y$-axis in first quadrant and find the area of triangle formed by points $P,O,Q$.

Find : \[ \int \frac{2x+1}{\sqrt{x^2+6x}}\,dx \]

If $\dfrac{d}{dx}(F(x))=\dfrac{1}{e^x+1}$, then find $F(x)$ given that $F(0)=\log\left(\dfrac{1}{2}\right)$.

$\displaystyle \int \frac{dx}{1+\cos x}$ is equal to

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 ____________.

মান নির্ণয় কর: \[ \int_{-1}^{1} e^{|x|}\, dx \]

কোন $ a $ ও $ b $-এর মানের জন্য নিম্নলিখিত গাণিতিক অভিব্যক্তিটি সত্য হবে? \[ \int \frac{dx}{1+\sin x} = \tan\left(\frac{x}{2} + a\right) + b \]

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 $m+n$.