The value of $\int_{e^2}^{e^4} \frac{1}{x} \left( \frac{e^{((\ln x)^2+1)^{-1}}}{e^{((\ln x)^2+1)^{-1}} + e^{((6-\ln x)^2+1)^{-1}}} \right) dx$ is

If $a=2n$ and $b=2m+1$ for all $m, n \in N$,then evaluate the integral: $\int_{-\pi}^{\pi} e^{\sin^a x} \cot^b((2n+1)x) dx$.

The integral $\int_0^\pi \frac{8 x \, dx}{4 \cos^2 x + \sin^2 x}$ is equal to

If $\alpha = 1$ and $\beta = 1 + i\sqrt{2}$, where $i = \sqrt{-1}$ are two roots of the equation $x^3 + ax^2 + bx + c = 0$ with $a, b, c \in R$, then $\int_{-1}^{1} (x^3 + ax^2 + bx + c) dx$ is equal to:

The value of $\int_{0}^{1} \tan ^{-1}\left(\frac{2 x-1}{1+x-x^{2}}\right) d x$ is

$\int_0^{\pi / 2} \frac{1}{1+\sqrt{\tan x}} d x=$