List of top Mathematics Questions on Integration by Partial Fractions asked in JEE Main

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 {an}ⁿ⁼⁰_∞be a sequence such that a₀=a₁=0 and a_n+2=3a_n+1−2a_n+1,∀ n≥0. Then a₂₅a₂₃−2a₂₅a₂₂−2a₂₃a₂₄+4a₂₂a₂₄is equal to

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:

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 the minimum value v₀ of
v = |z|²+|z-3|²+|z-6i|²,z∈C
is attained at z = z₀. Then
$|2z^2_0-\overline{z}^3_0+3|^2+v^2_0$
is equal to

Let m₁, m₂ be the slopes of two adjacent sides of a square of side a such that $a^2 + 11a + 3(m_1^2 + m_2^2) = 220$. If one vertex of the square is $(10(\cos \alpha - \sin \alpha), 10(\sin \alpha + \cos \alpha))$, where $\alpha \in \left(0, \frac{\pi}{2}\right)$and the equation of one diagonal is $(\cos \alpha - \sin \alpha)x + (\sin \alpha + \cos \alpha)y = 10$, then $72(\sin^4 \alpha + \cos^4 \alpha) + a^2 - 3a + 13$ is equal to :

If $\begin{array}{l}\int_{0}^{\sqrt{3}}\frac{15x^3}{\sqrt{1+x^2 + \sqrt{(1+x^2)^3}}}dx = \alpha \sqrt{2}+\beta\sqrt{3},\end{array}$where $α, β$ are integers, then $α + β$ is equal to

Let R₁ and R₂ be two relations defined on ℝ by a R₁b ⇔ ab ≥ 0 and aR₂b ⇔ a ≥ b. Then,

Let
f,g:N = {1} → N be functions defined by
f(a) = α, where α is the maximum of the powers of those primes p such that p^α divides a, and g(a) = a + 1, for all a ∈ N – {1}. Then, the function f + g is

Let
$I = ∫^{\frac{π}{3}}_{\frac{π}{4}}(\frac{8sinx-sin2x}{x})dx$
Then

Let y = y₁(x) and y = y₂(x) be two distinct solution of the differential equation
$\frac{dy}{dx} = x+y,$
with y₁(0) = 0 and y₂(0) = 1 respectively. Then, the number of points of intersection of y = y₁ (x) and y = y₂(x) is

Let P(a, b) be a point on the parabola y² = 8x such that the tangent at P passes through the centre of the circle x² + y² – 10x – 14y + 65 = 0. Let A be the product of all possible values of a and B be the product of all possible values of b. Then the value of A + B is equal to

$\sum_{r=1}^{20}$(r²+1)(r!) is equal to

For $I(x) = \int \frac{\sec^2 x - 2022}{\sin^{2022} x} \, dx, \quad \text{if } I\left(\frac{\pi}{4}\right) = 2^{1011}$, then

Let S={z=x+iy:|z–1+i|≥|z|,|z|<2,|z+i|=|z–1|}.
Then the set of all values of x, for which w = 2x + iy ∈ S for some y ∈ R is

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)