Embibe Experts Solutions for Chapter: Indefinite Integration, Exercise 3: EXERCISE-3

If $f\left(x\right)={\tan }^{-1}x+\ln \sqrt{1+x}-\ln \sqrt{1-x}$ and $\frac{1}{2}\int f^{\text{'}}\left(x\right)d\left(x^4\right)=-\ln \left(1-x^n\right)+C$, then the value of $n$ is

If $\int \sqrt{\frac{cosecx-\cot x}{cosecx+\cot x}}·\frac{\sec x}{\sqrt{1+2\sec x}}dx={\sin }^{-1}\left(\frac{1}{k}{\sec }^2\frac{x}{2}\right)+C$ then the value of $k$ is

If $\int \cos 2\theta ·\ln \left(\frac{\cos \theta +\sin \theta }{\cos \theta -\sin \theta }\right)d\theta =\frac{1}{\mathrm{k}}\sin 2\theta \left(\mathrm{lntan}\left(\frac{\pi }{4}+\theta \right)\right)-\frac{1}{\mathrm{k}}\ln \left(\sec 2\theta \right)$ then the value of $k$ is

If $\int \frac{x\ln \left(x\right)}{{\left(x^2-1\right)}^{3/2}}dx=f\left(x\right)+C$ and $f\left(\sqrt{2}\right)=\frac{3\pi }{k}-\frac{\ln 2}{2}$, then the value of $k$ is

If $\int {\left(\sin x\right)}^{-11/3}{\left(\cos x\right)}^{-1/3}dx=f\left(x\right)$ and $f\left(\frac{\pi }{2}\right)=0$, then $\left|4f\left(\frac{\pi }{4}\right)\right|$ is

Suppose $\int \frac{1-7{\cos }^{2}x}{{\sin }^{7}x{\cos }^{2}x}dx=\frac{g\left(x\right)}{{\sin }^{7}x}+C$, where $C$ is arbitrary constant of integration. Then find the value of $g^{\text{'}}\left(0\right)+g^{\text{'}\text{'}}\left(\frac{\pi }{4}\right)$

If $\int \frac{{\cos }^{6}xdx}{{\sin }^{4}x{\left({\sin }^{7}x+{\cos }^{7}x\right)}^{4/7}}=-\frac{1}{3}{\left(1+{\cot }^{A}x\right)}^B+C$, where $C$ is constant of integration, then $A^{2}B$ is equal to

If $\int \frac{2x^2+3x+3}{\sqrt{x^2+2x+2}}dx=ax\sqrt{x^2+2x+2}+b\ln \left|\left(x+1\right)+\sqrt{{\left(x+1\right)}^2+1}\right|+c$ ($c$ is integration constant) then $a+b$ is equal to