0-pi-2-1-1-sin-6-x-dx-

Question Number 167040 by cortano1 last updated on 05/Mar/22

\int_{0}^{\frac{π}{2}} \frac{1}{1 + \sin^{6} x} dx = ?

Commented by greogoury55 last updated on 05/Mar/22

I = \int_{0}^{\frac{π}{2}} \frac{\sec^{6} x}{\sec^{6} x + \tan^{6} x} d x

\overset{t = \tan x}{=} \int_{0}^{\infty} \frac{{(1 + t^{2})}^{2}}{{(1 + t^{2})}^{3} + t^{6}} d t

= \int_{0}^{\infty} \frac{t^{4} + 2 t^{2} + 1}{(2 t^{2} + 1) (t^{4} + t^{2} + 1)} d t

= \frac{1}{3} [\int_{0}^{\infty} \frac{d t}{2 t^{2} + 1} + \int_{0}^{\infty} \frac{t^{2} + 2}{t^{4} + t^{2} + 1} d t]

= \frac{π}{12} (\sqrt{2} + 2 \sqrt{3})