$\int_{-1}^1 \sin^5 x \cos^4 x \, dx = $ . . . . . .

If $\int_{-1}^4 f(x) dx = 4$ and $\int_2^4 \{3 - f(x)\} dx = 7$, then find the value of $\int_{-1}^2 f(x) dx$.

$\int_{ - \pi /2}^{\pi /2} {\log \left( {\frac{{2 - \sin \theta }}{{2 + \sin \theta }}} \right)\,d\theta = } $

$\int_{ - \pi /2}^{\pi /2} {\frac{{\sin x}}{{1 + {{\cos }^2}x}}{e^{ - {{\cos }^2}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

The value of the definite integral $\int_{0}^{\pi/2} \sqrt{\tan x} \, dx$ is: