Can't Figure Out This Integral? Let's Help Each Other!

[FeynmanIsCool]

Hello everyone,

I brushing up on integration techniques and I came across this problem in a book. Does anyone here know were to start? Even Wolfram blanked on it!

$\int_{0}^{\frac{\pi }{2}}\, \frac{1}{1+(tanx)^{\sqrt 2}} dx$

This integral appeared in the book before sequences and series, so the book did not intend to use any type of series expansion.

Let me know!

 

[Jorriss]

This is an integral from one of the old putnams, if you google around you can probably find it.

Consider the substitution of $$\frac{\pi}{2} - x$$ into $$f(x) = \frac{1}{1+tan^{\sqrt(2)}(x)}$$ This gives $$f(π/2 - x) = 1-f(x)$$ Now, break the integral up into two parts, one from 0 to pi/4 and the other from pi/4 to pi/2 and manipulate the second integral using the above identity.

 

[FeynmanIsCool]

Thanks for the reply,

You're right, I found the answer quickly after a google search. Very interesting integral!