Evaluate the integral:${\int }_{\frac{\pi }{3}}^{\frac{\pi }{2}}\frac{\sqrt{1+\cos x}}{(1-\cos x{)}^{{\displaystyle \frac{5}{2}}}}dx$.

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

Evaluate the integral:${\int }_0^{1}x{\left({\tan }^{-1}x\right)}^{2}dx$.

If ${\int }_0^1{\sin }^{-1}\sqrt{x}dx=\frac{\mathrm{\pi }}{k}$, then find the value of $k$.

Evaluate the integral:${\int }_0^a{\sin }^{-1}\sqrt{\frac{x}{a+x}}dx$.

Evaluate the integral:${\int }_0^9\frac{1}{(1+\sqrt{x})}dx$.

Evaluate the integral:${\int }_0^1{x}^3\sqrt{1+3x^4}dx$.

Evaluate the integral:${\int }_0^1\frac{\left(1-x^2\right)}{{\left(1+x^2\right)}^2}dx$.

Evaluate the integral:${\int }_1^2\frac{dx}{(x+1)\sqrt{x^2-1}}$.