Evaluate the definite integral $\int_0^4 x[x] \, dx$,where $[x]$ denotes the greatest integer function not greater than $x$.

If $\theta_{1}$ and $\theta_{2}$ are respectively the smallest and the largest values of $\theta$ in $(0, 2\pi) - \{\pi\}$ which satisfy the equation $2 \cot^{2} \theta - \frac{5}{\sin \theta} + 4 = 0$,then $\int_{\theta_{1}}^{\theta_{2}} \cos^{2} 3\theta \, d\theta$ is equal to

$A$ quadratic polynomial $P(x)$ satisfies the conditions $P(0) = 0$,$P(1) = 0$,and $\int_{0}^{1} P(x) dx = 1$. The leading coefficient of the quadratic polynomial is:

$\int_{-\pi / 4}^{\pi / 3}\left|\tan \left(x-\frac{\pi}{6}\right)\right| d x=$

Evaluate the definite integral: $\int_0^{\pi^2 / 4} (2 \sin \sqrt{x} + \sqrt{x} \cos \sqrt{x}) \, dx$

$\int_0^{2\pi } {\sqrt {1 + \sin \frac{x}{2}} \,dx = } $