How to move powers from denominator into numuerator? $\frac{x^n}{1+x^m}=\frac{x^{\frac{n-m+1}m}}{m(1+x)}$

Question

Here, robjohn simplifies $$\int_0^\infty\frac{x^n}{1+x^m}\mathrm{d}x =\frac1m\int_0^\infty\frac{x^{\frac{n-m+1}m}}{1+x}\mathrm{d}x$$ How do you go about this shuffling of powers into the numerator?

I've tried multiple factoring strategies such as factoring the bottom into $$x-e^{i\pi(2k+1)/3},\quad k=0,1,...,n-1$$ as well as attempting to simplify the end result back into the first, but to no avail.

A step by step demonstration would be really appreciated!

It's not a direct manipulation of the integrand. It's a u-substitution $x\rightarrow x^{1/m}$. — Christian E. Ramirez, Sep 27 '22 at 01:14
Substitution $y=x^m$, then re-name the dummy variable $y$ back to $x$. — GEdgar, Sep 27 '22 at 01:14

score 0 · Accepted Answer · answered Sep 27 '22 at 01:25

0

Substitution $y=x^m$, then re-name the dummy variable y back to x.

as per GEdgar's comment

answered Sep 27 '22 at 01:25

Max0815

3,505

How to move powers from denominator into numuerator? $\frac{x^n}{1+x^m}=\frac{x^{\frac{n-m+1}m}}{m(1+x)}$

1 Answers1