The value of $\int_{0}^{\sqrt{2}} [x^2] \, dx$,where $[.]$ denotes the greatest integer function.

Let $f(x) = \begin{cases} \frac{x}{\sin x}, & x \in (0, 1) \\ 1, & x = 0 \end{cases}$. Consider the integral $I_n = \sqrt{n} \int_0^{1/n} f(x) e^{-nx} dx$. Then,$\lim_{n \to \infty} I_n$ is:

$\int_{0}^{\frac{\pi}{2}} \frac{4x \sin x + x^2 \cos x}{2\sqrt{\sin x}} dx$ is equal to

$\int_0^1 x e^x \, dx = $ . . . . . .

If $\int_{n}^{n+1} g(x) dx = n^2, \forall n \in Z$,then the value of $\int_{-3}^3 g(x) dx$ is

Suppose that $F(x)$ is an antiderivative of $f(x) = \frac{\sin x}{x}$,$x > 0$. Then $\int_{1}^{3} \frac{\sin 2x}{x} dx$ can be expressed as: