Let $f(x) = \{x\}$ denote the fractional part of a real number $x$. Then, the value of $\int_{0}^{\sqrt{3}} f(x^2) dx$ is

The value of $\int_{-1}^{1} \frac{dx}{\sqrt{|x|}}$ is

$\mathop {Lim}\limits_{n \to \infty } \int_0^2 {\left( {1 + \frac{t}{{n + 1}}} \right)^n} dt$ is equal to

The minimum value of $\int_0^x {t{e^{ - {t^2}}}} dt$ is

Evaluate $\int_{-2}^1 f(x) dx$, where $f(x) = \begin{cases} 1-2x, & x \leq 0 \\ 1+2x, & x \geq 0 \end{cases}$

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