solve-dx-sin-10-x-cos-10-x-

Question Number 195602 by mokys last updated on 05/Aug/23

s o l v e \int \frac{d x}{{s i n}^{10} (x) + {c o s}^{10} (x)}

Answered by Frix last updated on 06/Aug/23

\int \frac{d x}{\sin^{10} x + \cos^{10} x} =

= 64 \int \frac{d x}{{5 \cos}^{2} 4 x + 30 \cos 4 x + 29} =

= \frac{5}{\sqrt{5}} \int \frac{d x}{3 - \frac{4}{\sqrt{5}} + \cos 4 x} - \frac{8}{\sqrt{5}} \int \frac{d x}{3 + \frac{4}{\sqrt{5}} + \cos 4 x} =

[using t = \tan 2 x]

= (4 + 2 \sqrt{5}) \int \frac{d t}{t^{2} + 6 + 2 \sqrt{5}} + (4 - 2 \sqrt{5}) \int \frac{d t}{t^{2} + 6 - 2 \sqrt{5}} =

= - (\frac{3 + \sqrt{5}}{2} \tan^{- 1} \frac{(1 - \sqrt{5}) t}{4} + \frac{3 - \sqrt{5}}{2} \tan^{- 1} \frac{(1 + \sqrt{5}) t}{4}) =

= - (\frac{3 + \sqrt{5}}{2} \tan^{- 1} \frac{(1 - \sqrt{5}) \tan 2 x}{4} + \frac{3 - \sqrt{5}}{2} \tan^{- 1} \frac{(1 + \sqrt{5}) \tan 2 x}{4}) + C

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