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:

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

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

Evaluate  ${\displaystyle \int \frac{x^2}{x^2+6x+12}}dx.$

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

Let $f$ be a positive function.

Let  $I_1={\displaystyle \underset{\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}1-k}{\overset{k}{\int }}xf[x(\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}1-x)]}dx,I_2={\displaystyle \underset{\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}1-k}{\overset{k}{\int }}f[x(\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}1-x)]}dx,$  where  $2k-1>0.$  then  $\frac{I_1}{I_2}$  is

For any natural number m, evaluate

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

Evaluate the following integral $\int \frac{x^2}{\sqrt{1-x}}dx$

Evaluate the following

 ${\displaystyle \int \frac{dx}{x^2{(x^4+1)}^{\frac{3}{4}}}}$