The value of $\int \limits_{0}^{\pi} \frac{e^{\cos x} \sin x}{\left(1+\cos ^{2} x\right)\left(e^{\cos x}+e^{-\cos x}\right)} d x$ is equal to

$\int_0^{\pi /4} {\log (1 + \tan \theta )\,d\theta = } $

If $I = \int\limits_0^{\frac{\pi}{2}} \ln(\sin x) dx$,then $\int\limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \ln(\sin x + \cos x) dx =$

Let $h(x) = \int\limits_0^x {g(t)dt}$,where $g(x)$ is a differentiable and odd function $\forall x \in R$ and $g(x)$ is periodic with period $3$.
Statement $1: h(x) + h(-x) = 0$ $\forall x \in R$
Statement $2: h(x) + h(-x) = 2 \int\limits_0^x {g(t)dt}$ $\forall x \in R$
Statement $3: h(3n) = 0$ $\forall n \in I$
Then which of the following statement$(s)$ is/are true?

Let $f(x)=\int_0^x g(t) \log _e\left(\frac{1-t}{1+t}\right) d t$,where $g$ is a continuous odd function. If $\int_{-\pi / 2}^{\pi / 2}\left(f(x)+\frac{x^2 \cos x}{1+e^x}\right) d x=\left(\frac{\pi}{\alpha}\right)^2-\alpha$,then $\alpha$ is equal to..............

$\int_{-8}^{8} \frac{x^{5}+x^{3}}{4-x^{2}} \, dx = $