Evaluate $\int_{-1}^{1} \sin^{5} x \cos^{4} x \, dx$

Let $p(x)$ be a function defined on $R$ such that $p'(x) = p'(1 - x)$ for all $x \in [0, 1]$,$p(0) = 1$,and $p(1) = 41$. Then $\int_{0}^{1} p(x) dx = $

$\int_{0}^{2a} f(x) \, dx = $

If $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \sqrt{1-\sin 2x} \, dx = \alpha + \beta \sqrt{2} + \gamma \sqrt{3}$,where $\alpha, \beta$ and $\gamma$ are rational numbers,then $3\alpha + 4\beta - \gamma$ is equal to ..........

By using the properties of definite integrals,evaluate the integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \sin ^{2} x \, dx$.

$\int_{0}^{\pi} \frac{\cos ^{4} x}{\cos ^{4} x+\sin ^{4} x} d x$ is equal to