Embibe Experts Solutions for Chapter: Indefinite Integration, Exercise 1: JEE Main - 10th January 2019 Shift 2

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

Let $g:\left(0,\infty \right)\to \mathit{R}$ be a differentiable function such that $\int \left(\frac{x\left(\cos x-\sin x\right)}{{\mathrm{e}}^x+1}+\frac{g\left(x\right)\left({\mathrm{e}}^x+1-x{\mathrm{e}}^x\right)}{{\left({\mathrm{e}}^x+1\right)}^2}\right)\mathrm{d}x=\frac{xg\left(x\right)}{{\mathrm{e}}^x+1}+C$, for all $x>0$, where $C$ is an arbitrary constant. Then

If $\int \frac{1}{x}\sqrt{\frac{1-x}{1+x}}dx=g\left(x\right)+c,g\left(1\right)=0$, then $g\left(\frac{1}{2}\right)$ is equal to

$\int \frac{\left(x^2+1\right)e^x}{{\left(x+1\right)}^2}dx=f\left(x\right)e^x+C$, where $C$ is a constant, then $\frac{d^{3}f}{d{x}^3}$ at $x=1$ is equal to

The integral $\int \frac{\left(1-\frac{1}{\sqrt{3}}\right)\left(\cos x-\sin x\right)}{\left(1+\frac{2}{\sqrt{3}}\sin 2x\right)}dx$ is equal to

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

Let $f\left(x\right)=\int \frac{2x}{\left(x^2+1\right)\left(x^2+3\right)}dx$. If $f\left(3\right)=\frac{1}{2}\left({\log }_{e}5-{\log }_{e}6\right)$, then $f\left(4\right)$ is equal to

If $\int \sqrt{\sec 2x-1}dx=\alpha {\log }_e\left|\cos 2x+\beta +\sqrt{\cos 2x\left(1+\cos \frac{1}{\beta }x\right)}\right|$   $+$ constant, then $\beta -\alpha$ is equal to ______.