TITLE: Finding $\int\frac{1}{x^{11}+4x^6}dx$
QUESTION [8 upvotes]: I wanted to find out if there is an easy way to evaluate $$\displaystyle\int\frac{1}{x^{11}+4x^6}dx.$$
I substituted $u=x^5$ and then used partial fractions, but maybe there is a simpler way to find this.

REPLY [5 votes]: Let $t=\frac{1}{x}$, so then $x=\frac{1}{t}$ and $dx=-\frac{1}{t^2}dt$.
Then $\displaystyle\int\frac{1}{x^{11}+4x^6}dx=\int\frac{1}{\frac{1}{t^{11}}+\frac{4}{t^6}}\left(-\frac{1}{t^2}\right)dt=-\int\frac{t^9}{1+4t^5}dt$
$=-\int\big(\frac{1}{4}t^4-\frac{\frac{1}{4}t^4}{1+4t^5}\big)dt=-\frac{1}{20}t^5+\frac{1}{80}\ln\vert1+4t^5\vert+C=-\frac{1}{20x^5}+\frac{1}{80}\ln\vert1+\frac{4}{x^5}\vert+C$