List of top Mathematics Questions on Integration by Partial Fractions

If \[ \int (\sin x)^{-\frac{11}{2}} (\cos x)^{-\frac{5}{2}} \, dx \] is equal to \[ -\frac{p_1}{q_1}(\cot x)^{\frac{9}{2}} -\frac{p_2}{q_2}(\cot x)^{\frac{5}{2}} -\frac{p_3}{q_3}(\cot x)^{\frac{1}{2}} +\frac{p_4}{q_4}(\cot x)^{-\frac{3}{2}} + C, \] where $ p_i, q_i $ are positive integers with $ \gcd(p_i,q_i)=1 $ for $ i=1,2,3,4 $, then the value of \[ \frac{15\,p_1 p_2 p_3 p_4}{q_1 q_2 q_3 q_4} \] is ___________.

Let for $ f(x) = 7\tan^8 x + 7\tan^6 x - 3\tan^4 x - 3\tan^2 x $, $ I_1 = \int_0^{\frac{\pi}{4}} f(x)dx $ and $ I_2 = \int_0^{\frac{\pi}{4}} x f(x)dx $. Then $ 7I_1 + 12I_2 $ is equal to:

Evaluate the integral $\int \frac{\pi}{x^n + 1 - x} , dx$:

$\text{Evaluate } \int e^x \left( \frac{2x + 1}{2 \sqrt{x}} \right) dx:$

The value of $\frac {120}{\pi^3}|∫_0^\pi\frac {x^2sinx.cosx}{(sinx)^4+(cosx)^4}dx|$ is

The integral $$ \int \frac{x^8 - x^2}{(x^{12} + 3x^6 + 1) \tan^{-1}\left( \frac{x^3 + 1}{x^3} \right)} \, dx $$ is equal to:

$\displaystyle\sum_{k=0}^6{ }^{51-k} C_3$ is equal to

Let
A = $\begin{pmatrix}1 & 2\\ -2 & -5 \end{pmatrix}$
Let $ \alpha, \beta \in \mathbb{R} $ be such that $ \alpha A^2 + \beta A = 2I $. Then $ \alpha + \beta $ is equal to

Suppose a₁, a₂, … a_n, … be an arithmetic progression of natural numbers. If the ration of the sum of first five terms to the sum of first nine terms of the progression is 5 : 17 and 110 < a₁₅ < 120, then the sum of the first ten terms of the progression is equal to

$lim_{n→∞} \frac1{2^n}(\frac1{\sqrt{1-1/2^n}} +\frac1{\sqrt{1-\frac2{2^n}}}+\frac1{\sqrt{1-\frac3{2^n}}}+...+\frac1{\sqrt{1-\frac{2^n-1}{2^n}}})$ is equal to

If A and B are two events such that $P(A)=\frac1{3},P(B)=\frac1{5}$ and $P(A∪B)=\frac1{2}$ then $P(\frac{A}{B'}) + P(\frac{B}{A'})$ is equal to

A common tangent T to the curves
$C_1:\frac{x^2}{4}+\frac{y^2}{9} = 1$
and
$C_2:\frac{x^2}{4^2}\frac{-y^2}{143} = 1$
does not pass through the fourth quadrant. If T touches C₁ at (x₁, y₁) and C₂ at (x₂, y₂), then |2x₁ + x₂| is equal to ______.

2sin($\frac{\pi}{22}$)sin($\frac{3\pi}{22}$)sin($\frac{5\pi}{22}$)sin($\frac{7\pi}{22}$)sin($\frac{9\pi}{22}$) is equal to

If (2, 3, 9), (5, 2, 1), (1, λ, 8) and (λ, 2, 3) are coplanar, then the product of all possible values of λ is :

The value of the integral ∫ sinθ sin2θ (sin⁶θ + sin⁴θ + sin²θ) / √(2sin⁴θ + 3sin²θ + 6) dθ is: (where c is a constant of integration)

If $\int \frac{\cos x - \sin x}{\sqrt{8 - \sin 2x}} dx = a \sin^{-1} \left( \frac{\sin x + \cos x}{b} \right) + c$, find (a, b) :