Mathematics : Integrals

Evaluate: $\int {\mathrm{e}}^{\mathrm{x}} \left( \frac{\sin 4\mathrm{x} - 4}{1 - \cos 4\mathrm{x}}\right) \mathrm{dx}$

Evaluate: $\int \frac{1 - {\mathrm{x}}^2}{\mathrm{x} \left( 1 - 2\mathrm{x} \right)} \mathrm{dx}$

Evaluate ${\int }_1^3 \left( 3 {\mathrm{x}}^2 + 2 \mathrm{x} \right) \mathrm{dx}$  as limit of sums.

Write the value of $\int \sec \mathrm{x} \left( \sec \mathrm{x} + \tan \mathrm{x} \right) \mathrm{dx}$

Write the value of $\int \frac{\mathrm{dx}}{{\mathrm{x}}^2 + 16}$

Evaluate: $\int \frac{5 \mathrm{x} + 3}{\sqrt{ {\mathrm{x}}^2 + 4 \mathrm{x} + 10}} \mathrm{dx}$

Evaluate: $\int \frac{2\mathrm{x}}{\sqrt{ \left( {\mathrm{x}}^2 + 1 \right) \left( {\mathrm{x}}^2 + 3 \right)}} \mathrm{dx}$

Evaluate: ${\int }_0^{\frac{\mathrm{\pi }}{2}} 2 \sin \mathrm{x} \cos \mathrm{x} {\tan }^{-1} \left( \sin \mathrm{x} \right) \mathrm{dx}$

Evaluate: ${\int }_0^{\frac{\mathrm{\pi }}{2}} \frac{\mathrm{x} \sin \mathrm{x} \cos \mathrm{x}}{{\sin }^4 \mathrm{x} + {\cos }^4 \mathrm{x}} \mathrm{dx}$

Evaluate $\int ( 1 - \mathrm{x} ) \sqrt{ \mathrm{x}} \mathrm{dx}$