The value of $\int_{-2}^{2}(a x^{3}+b x+c) d x$ depends on the

$\int_{\pi / 6}^{\pi / 3} \frac{1}{1+\sqrt{\cot x}} d x=$

By using the properties of definite integrals,evaluate the integral $\int_{0}^{\frac{\pi}{2}} \frac{\sin x-\cos x}{1+\sin x \cos x} d x$.

$\int\limits_0^\pi (x \cdot \sin^2 x \cdot \cos x) dx =$

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?

$\int_0^\pi \sin^2 x \, dx$ is equal to