If $\int_0^1 \frac{1}{\sqrt{3+x}+\sqrt{1+x}} d x=a+b \sqrt{2}+c \sqrt{3}$,where $a, b, c$ are rational numbers,then $2 a+3 b-4 c$ is equal to :

If $g(x) = \int_{0}^{x} \cos 4t \, dt$,then $g(x + \pi) = $

The integral $\int_0^1 \cot^{-1}(1 + x + x^2) dx$ is equal to:

If the value of the integral $\int_{0}^{5} \frac{x+[x]}{e^{x-[x]}} \,dx = \alpha e^{-1} + \beta$,where $\alpha, \beta \in R, 5\alpha + 6\beta = 0$,and $[x]$ denotes the greatest integer less than or equal to $x$; then the value of $(\alpha + \beta)^{2}$ is equal to:

$\left[ \int_{0}^{2} \sqrt{x + \sqrt{x + \sqrt{x + \dots \infty}}} \, dx \right]$ is equal to (where $[\cdot]$ is the $G.I.F.$)

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