$\int_0^{\frac{\pi}{2}} \frac{\sin^2 x}{\sin x + \cos x} dx = $

Let $f$ and $g$ be continuous functions on $[0, a]$ such that $f(x)=f(a-x)$ and $g(x)+g(a-x)=4$,then $\int_0^a f(x) g(x) d x$ is equal to

If $f(x) = \int_0^{\sin^2 x} \sin^{-1} \sqrt{t} \, dt$ and $g(x) = \int_0^{\cos^2 x} \cos^{-1} \sqrt{t} \, dt$, then the value of $f(x) + g(x)$ is

Let $f(x)$ be a differentiable function defined on $[0,2]$ such that $f^{\prime}(x) = f^{\prime}(2-x)$ for all $x \in (0,2)$,$f(0) = 1$ and $f(2) = e^{2}$. Then the value of $\int_{0}^{2} f(x) dx$ is ..... .

$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \log \left(\frac{2-\sin \theta}{2+\sin \theta}\right) d \theta$ is equal to

The value of the integral $\int_{-\pi}^{\pi} \frac{\cos^2 x}{1+a^x} dx$,where $a > 0$,is