Evaluate:

${\int }_e^{2e}\left\{\frac{1}{\log x}-\frac{1}{(\log x{)}^2}\right\}dx$

Evaluate:

${\int }_1^{e^2}\frac{\log x}{(1+\log x{)}^2}dx$

Evaluate :

${\int }_0^{\frac{\pi }{2}}\frac{\sin x+\cos x}{9+16\sin 2x}dx$

Evaluate :

${\int }_{\frac{\pi }{2}}^{\pi }\frac{\sqrt{1+\cos x}}{(1-\cos x{)}^{\frac{5}{2}}}dx$

If ${\int }_0^1{\left({\cos }^{-1}x\right)}^{2}dx =\mathrm{\pi }-\mathrm{k}$, then the value of $k$ is

Find the value of $k$. If${\int }_0^{\frac{\pi }{2}}(\sqrt{\tan x}+\sqrt{\cot x})dx=\mathrm{\pi }\sqrt{\mathrm{k}}$.

Evaluate:

${\int }_0^{\frac{\pi }{2}}{e}^{\frac{x}{2}}\sin \left(\frac{\pi }{2}+\frac{x}{2}\right)dx$

Evaluate:

${\int }_0^1\left(x{e}^x+\sin \frac{\pi x}{4}\right)dx$