Putnam Definite Integral Hard Problem: Evaluate $\displaystyle\int_0^{\frac{\pi}{2}}\frac{dx}{1+(\tan x)^{\sqrt{2}}}$.

Another hard definite integral but it is not as bad as it looks. Let's first try the substitution method:

Attempt I:

We are given to evaluate:

[MATH]I=\int_0^{\pi/2}\frac{1}{1+\left(\tan(x)\right)^{\sqrt{2}}}\,dx[/MATH]

If we use the substitution:

[MATH]u=\frac{\pi}{2}-x[/MATH]