How to find $ \displaystyle \int_0^{\pi/2} \frac{1}{1+(\tan x)^e}dx $?

Question

How to find $$\displaystyle \int_0^{\pi/2} \frac{1}{1+(\tan x)^e}dx $$Substitution seems not to work.

Note for close voters: The question linked has different parameters, but the accepted answer is general. Also just in case anyone is interested, I think this question originally comes from a putnam. — Peter Woolfitt, Apr 01 '17 at 15:30
Ha, things become surprisingly easy after we do the right substitution and the $I+I$ trick! Thanks! — Hank, Apr 02 '17 at 02:27

score 1 · Accepted Answer · answered Apr 01 '17 at 15:23

1

Hint:

$$I= \int_0^{\pi/2} \frac{1}{1+(\tan x)^e}dx$$

$$= \int_0^{\pi/2} \frac{1}{1+(\tan (\frac{\pi}{2}-x))^e}dx$$

$$=\int_0^{\pi/2} \frac{(\tan(x))^e}{1+(\tan x)^e}dx$$

answered Apr 01 '17 at 15:23

jonsno

1 Answers1