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If $\int_{0}^{1} \frac{1}{\sqrt{3+x}+\sqrt{1+x}} \, dx = a + b\sqrt{2} + c\sqrt{3},$ where $a, b, c$ are rational numbers, then $2a + 3b - 4c$ is equal to:

If \(\displaystyle \int \frac{1}{\sqrt[5]{(x-1)^4 (x+3)^6}} \, dx = A \left( \frac{\alpha x - 1}{\beta x + 3} \right)^B + C \), where \( C \) is the constant of integration, then the value of \( \alpha + \beta + 20AB \) is __________.

For \( x \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \), if \( y(x) = \int \frac{cosec x + \sin x}{cosec x \sec x + \tan x \sin^2 x} \, dx \), and \( \displaystyle \lim_{x \to \left( \frac{\pi}{2} \right)^{-}} y(x) = 0 \) then \( y\left( \frac{\pi}{4} \right) \) is equal to:

Let \( \int \frac{2-\tan x}{3+\tan x}\,dx=\frac{1}{2}\left(\alpha x+\log_{e}\left| \beta\sin x+\gamma\cos x \right|\right)+C \), where \( C \) is the constant of integration. Then \( \alpha+\frac{\gamma}{\beta} \) is equal to:

Let \( I(x) = \int \frac{6}{\sin^2 x \, (1-\cot x)^2} \, dx \). If \( I(0) = 3 \), then \( I\!\left(\frac{\pi}{12}\right) \) is equal to:

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

If \( \displaystyle \int cosec^{5}x\,dx=\alpha \cot x \ x\ cosec\ x\left(cosec^{2}x+\tfrac{3}{2}\right)+\beta \log_{e}\left|\tan \tfrac{x}{2}\right|+C \) where \( \alpha,\beta\in\mathbb{R} \) and \( C \) is the constant of integration, then the value of \( 8{(\alpha+\beta)} \) equals __________.

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JEE Main 2024 9th April Morning Shift

Let \( \int \frac{2-\tan x}{3+\tan x}\,dx=\frac{1}{2}\left(\alpha x+\log_{e}\left| \beta\sin x+\gamma\cos x \right|\right)+C \), where \( C \) is the constant of integration. Then \( \alpha+\frac{\gamma}{\beta} \) is equal to: