The value of the integral $\sum\limits_{k = 1}^n {\int_0^1 {f(k - 1 + x)\,dx} } $ is

$\int_0^{\pi} x f(\sin x) \, dx$ is equal to

$\int_0^{\pi / 2} \frac{16 x \sin x \cos x}{\sin ^4 x+\cos ^4 x} d x$ 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

Evaluate $\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{d x}{1+\sqrt{\tan x}}$

If $I = \int_0^\pi x \left\{ \sin^2(\sin x) + \cos^2(\cos x) \right\} dx$,then $[I] = \ldots$. Here,$[.]$ denotes the greatest integer function.