$\int_0^{1000} e^{x-[x]} \, dx$ is equal to

If $f$ and $g$ are continuous functions in $[0, a]$ satisfying $f(x) = f(a - x)$ and $g(x) + g(a - x) = 4$,then $\int_{0}^{a} f(x) g(x) dx$ is equal to:

The value of $\int_0^\pi \sin^{50} x \cos^{49} x \, dx$ is

$\int_0^{\frac{\pi}{2}} \frac{\sum_{n=0}^4 \left(\frac{n \pi}{4}+x\right)}{\cos x+\sin x} d x=$

The value of $\int_{0}^{2 \pi} \frac{x \sin^{8} x}{\sin^{8} x + \cos^{8} x} dx$ is equal to

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: