The value of $\int_{-1}^{3} \left( \tan^{-1} \left( \frac{x}{x^2+1} \right) + \tan^{-1} \left( \frac{x^2+1}{x} \right) \right) dx$ is

$\int_{\frac{\pi}{18}}^{\frac{4\pi}{9}} \frac{2\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx = . . . . . .$

If $\int_{0}^{1} \tan ^{-1} x \, dx = p$,then the value of $\int_{0}^{1} \tan ^{-1}\left(\frac{1-x}{1+x}\right) \, dx$ is

If $\int_{-1}^4 f(x) dx = 4$ and $\int_2^4 (3 - f(x)) dx = 7$,then $\int_{-1}^2 f(x) dx = $

Let $f(x) = \int_{0}^{x} g(t) dt$,where $g$ is a non-zero even function. If $f(x+5) = g(x)$,then $\int_{0}^{x} f(t) dt$ 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?