$\int_{-1}^1 \left(\sqrt{1+x+x^2}-\sqrt{1-x+x^2}\right) dx =$

$\int_{\frac{1}{2}}^2 \frac{1}{x} \operatorname{cosec}^{101}\left(x-\frac{1}{x}\right) d x=$

If $f(x) = \begin{cases} e^{\cos x}\sin x, & |x| \le 2 \\ 2, & \text{otherwise} \end{cases}$,then $\int_{-2}^{3} f(x) dx$ is equal to

The value of the integral $\int_0^{\frac{\pi}{4}} \frac{x \, dx}{\sin^4(2x) + \cos^4(2x)}$ equals :

If $m, l, r, s, n$ are integers such that $9 > m > l > s > n > r > 2$ and $\int_{-2 \pi}^{2 \pi} \sin ^m x \cos ^n x \, dx = 4 \int_0^\pi \sin ^m x \cos ^n x \, dx$, $\int_{-\pi}^\pi \sin ^r x \cos ^s x \, dx = 4 \int_0^{\pi / 2} \sin ^r x \cos ^s x \, dx$ and $\int_{-\pi / 2}^{\pi / 2} \sin ^l x \cos ^m x \, dx = 0$, then which of the following is true?

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