TITLE: Why is $\int\limits_0^1 (1-x^7)^{1/5} - (1-x^5)^{1/7} dx=0$?
QUESTION [18 upvotes]: When I tried to approximate $$\int_{0}^{1} (1-x^7)^{1/5}-(1-x^5)^{1/7}\ dx$$ I kept getting answers that were really close to $0$, so I think it might be true. But why? When I ask Mathematica, I get a bunch of symbols I don't understand!

REPLY [7 votes]: $\int_0^1(1-x^m)^{(1/n)}dx=(m+n)\Gamma(1/m)\Gamma(1/n)/\Gamma(1/m+1/n)$ is symmetric in $m, n$.