Evaluate $\underset{0}{\overset{1}{\int }}x{\left(1-x\right)}^5 dx$

By using properties of definite integrals, evaluate $\underset{0}{\overset{\raisebox{1ex}{$\text{π}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\int }}\frac{{\sin }^{2}x}{1+\sin x\cos x}dx$.

Evaluate $\underset{0}{\overset{1}{\int }}{\tan }^{-1}\left(\frac{2x-1}{1+x-x^2}\right)dx$

By using properties of definite integrals, evaluate$\underset{0}{\overset{\raisebox{1ex}{$\text{π}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\int }}\frac{a sec x + b \cos ec x}{sec x + \cos ec x}dx.$