$\int_{0}^{\pi} e^{\sin^2 x} \cos^3 x \, dx$ is equal to

$\int_0^{\alpha / 3} \frac{f(x)}{f(x)+f\left(\frac{\alpha-3 x}{3}\right)} d x=$

For any real number $x$,let $[x]$ denote the greatest integer less than or equal to $x$. Let $f$ be a real-valued function defined on the interval $[-10, 10]$ by $f(x) = \begin{cases} x - [x], & \text{if } [x] \text{ is odd} \\ 1 + [x] - x, & \text{if } [x] \text{ is even} \end{cases}$. Then the value of $\frac{\pi^2}{10} \int_{-10}^{10} f(x) \cos(\pi x) dx$ is:

$\int_{-1}^{1} \sin^{11} x \, dx$ is equal to

$\int\limits_0^{\pi / 2n} \frac{dx}{1 + \tan^n(nx)} = $

The value of the integral $\int_{-2}^{2} \frac{|x^{3}+x|}{e^{x|x|}+1} dx$ is equal to