$\int \cos^{-3/7} x \sin^{-11/7} x \, dx = $

If $\int \frac{1}{x^4+8 x^2+9} d x = \frac{1}{k} \left[ \frac{1}{\sqrt{14}} \tan^{-1}(f(x)) - \frac{1}{\sqrt{2}} \tan^{-1}(g(x)) \right] + c$, then $\sqrt{\frac{k}{2} + f(\sqrt{3}) + g(1)} =$

If $\int {\frac{1}{{x + {x^5}}}dx = f(x) + c} $,then the value of $\int {\frac{{{x^4}}}{{x + {x^5}}}dx} $ is

For any integer $n \geq 2$,if $I_n = \int \cot^n x \, dx$,then $I_5 =$

$\int \sqrt{\frac{1 - \sqrt{x}}{1 + \sqrt{x}}} \, dx = $

If $\int e^x \cos x \, dx = \frac{e^x}{2}(\cos x + \sin x)$ and $\int \frac{\cos \left(\log \left(\frac{2x+3}{3-2x}\right)\right)}{(3-2x)^2} \, dx = \frac{f(x)}{24}[\cos (g(x)) + \sin (g(x))] + c$, then $g(1) =$