0-pi-2-dx-1-tan-5-x-

Question Number 151423 by peter frank last updated on 21/Aug/21

\int_{0}^{\frac{π}{2}} \frac{dx}{1 + \tan^{5} x}

Answered by Olaf_Thorendsen last updated on 21/Aug/21

I = \int_{0}^{\frac{π}{2}} \frac{d x}{1 + \tan^{5} x}

I = \int_{0}^{\frac{π}{2}} \frac{\cos^{5} x}{\cos^{5} x + \sin^{5} x} d x (1)

Let u = \frac{π}{2} - x :

I = \int_{0}^{\frac{π}{2}} \frac{\sin^{5} u}{\sin^{5} u + \cos^{5} u} d u (2)

\frac{(1) + (2)}{2} : I = \frac{π}{4}