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JEE Mains | PYQ

The integral $\int \frac{1}{\sqrt[4]{(x-1)^3 (x+2)^5}} \, dx$ is equal to :
(Where $C$ is a constant of integration)

If \( f:\mathbb{R}\to\mathbb{R} \) is given by \( f(x)=x+1 \), then the value of \( \displaystyle \lim_{n \to \infty} \dfrac{1}{4} \left[ f(0) + f\left(\dfrac{5}{n}\right) + f\left(\dfrac{10}{n}\right) + ... + f\left(\dfrac{5(n - 1)}{n}\right) \right],\) is:

If \( \displaystyle \lim_{x \to 0} \frac{\sin^{-1}x - \tan^{-1}x}{3x^3} \) is equal to \( L \), then the value of \( (6L + 1) \) is:

The integral \( \int \frac{(2x-1)\cos\!\left(\sqrt{(2x-1)^2+5}\right)}{\sqrt{4x^2-4x+6}}\,dx \) is equal to:
(where \( c \) is a constant of integration)

If \( \int \frac{2e^{x} + 3e^{-x}}{4e^{x} + 7e^{-x}} \, dx = \frac{1}{14}\left( ux + v \log_{e}(4e^{x} + 7e^{-x}) \right) + C \), where \( C \) is a constant of integration, then \( u + v \) is equal to ________.

The function \( f(x)=\left|x^{2}-2x-3\right|\cdot e^{\left|9x^{2}-12x+4\right|} \) is not differentiable at exactly:

If \( \int \dfrac{dx}{(x^2 + x + 1)^2} = a \tan^{-1}\!\left( \dfrac{2x + 1}{\sqrt{3}} \right) + b \left( \dfrac{2x + 1}{x^2 + x + 1} \right) + C,\ x > 0 \) where \( C \) is the constant of integration, then the value of \( 9(\sqrt{3}\,a + b) \) is equal to __________.

If \( f(x) = \int \frac{5x^8 + 7x^6}{(x^2 + 1 + 2x^7)^2} \, dx , \; (x \geq 0), \; f(0) = 0 \) and \( f(1) = \frac{1}{k} \), then the value of \( K \) is ____________.

\( If \int \dfrac{\sin x}{\sin^{3}x+\cos^{3}x}\,dx=\alpha \log_{e}\!\left|1+\tan x\right|+\beta \log_{e}\!\left|1-\tan x+\tan^{2}x\right|+\gamma \tan^{-1}\!\left(\dfrac{2\tan x-1}{\sqrt{3}}\right)+C \), when \( C \) is constant of integration, then the value of \( 18(\alpha+\beta+\gamma^{2}) \) is ______.