Methods of Integration

Evaluate: $\int \frac{e^{{\tan }^{-1}x}}{1+x^2}dx$

Evaluate :  ${\displaystyle \int \frac{6x+7}{\sqrt{(x-5)(x-4)}}}dx$

 ${\displaystyle \int \frac{dx}{\sqrt{5-4x-2x^2}}}$  is equal to:

 ${\displaystyle \int \frac{\cos \sqrt{x}}{\sqrt{x}}}dx$ is equal to:

 ${\displaystyle \int \frac{\sin \sqrt{x}}{\sqrt{x}}dx}$  is equal to

Evaluate ${\displaystyle \int \sin 7x\sin xdx}$

The value of  ${\displaystyle \int \sqrt{\frac{1-x}{1+x}}dx}$ would be

Evaluate:  $\int \frac{\sin 2x}{{(a+b\cos x)}^2}dx$

Given, $\int e^x(\tan x+1)\sec xdx=e^{x}f\left(x\right)+C$

A satisfying value of  $f\left(x\right)$  would be:

What would be the value of  $\int \frac{x {\sin }^{-1}x}{\sqrt{1-x^2}}dx$

If $I=\int \frac{dx}{x^3+x^2+x+1}.$ then value of $I$ is equal to

Evaluate  ${\displaystyle \int \frac{(x+1)}{x{(1+x{e}^x)}^2}dx}$

For any natural number m, evaluate

 ${\displaystyle \int (x^{\mathrm{}\mathrm{}\mathrm{}\mathrm{}3m}+x^{\mathrm{}\mathrm{}\mathrm{}\mathrm{}2m}+x^m){(2x^{\mathrm{}\mathrm{}\mathrm{}\mathrm{}2m}+3x^m+6)}^{\frac{1}{m}}dx,x>0}$