Why is $\int\limits_0^1 (1-x^7)^{1/5} - (1-x^5)^{1/7} dx=0$?

When I attempted to approximate $$\int_{0}^{1} (1-x^7)^{1/5}-(1-x^5)^{1/7}\ dx$$ I maintained getting the answer that were actually near $0$, so I assume it could be real. Yet why? When I ask Mathematica, I get a number of icons I do not recognize!

2019-05-06 21:46:11
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Answers: 1

Note that if

$$ y = \left(1 - x^7\right)^{1/5} $$

after that

$$ \left(1 - y^5\right)^{1/7} = x $$

This suggests $(1-x^7)^{1/5}$ is the inverted function of $(1-x^5)^{1/7}$. In the chart, one will certainly coincide as the various other when mirrored along the angled line y = x.

Also, both features

  1. share the very same array [0, 1 ] and also domain name [0, 1 ] and also
  2. monotonically lowering,

Therefore, the location under the chart in [0, 1 ] will certainly coincide for both features :

$$ \int_0^1 \left(1-x^7\right)^{1/5} dx = \int_0^1 \left(1-y^5\right)^{1/7} dy $$

Grouping both integrals produce the formula in the title.

2019-05-08 19:52:44