If $f(x) = \int \frac{5x^{8} + 7x^{6}}{(x^{2} + 1 + 2x^{7})^{2}} dx$,$(x \geq 0)$,$f(0) = 0$ and $f(1) = \frac{1}{K}$,then the value of $K$ is

$\int {\frac{1}{{{x^2} - 1}}} \,\ln \left( {\frac{{x - 1}}{{x + 1}}} \right)dx$ equals :

If $n \geq 2$ is a natural number and $0 < \theta < \frac{\pi}{2}$,then $\int \frac{(\cos ^n \theta-\cos \theta)^{1 / n}}{\cos ^{n+1} \theta} \sin \theta d \theta =$

If $f(x)=\int \frac{x^2 \, dx}{(1+x^2)(1+\sqrt{1+x^2})}$ and $f(0)=0$,then $f(1)$ is

$\int \frac{4 x^2 \cot ^{-1}\left(x^3\right)}{1+x^6} \,d x=$ (where $C$ is a constant of integration.)

Find the value of $k$ if $\int \cos ^k(x) \sin (x) d x = \frac{-1}{4} \cos ^4(x) + C$.