Answer
$$\frac{1}{5}\ln \left| {1 + {x^5}} \right| + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{{x^4}}}{{1 + {x^5}}}} dx \cr
& {\text{Set }}u = 1 + {x^5},{\text{ }}du = 5{x^4}dx \to {x^4}dx = \frac{1}{5}du \cr
& {\text{using the indicated substitution}} \cr
& \int {\frac{{{x^4}}}{{1 + {x^5}}}} dx = \int {\frac{{\left( {1/5} \right)du}}{u}} \cr
& = \frac{1}{5}\int {\frac{{du}}{u}} \cr
& {\text{integrate}} \cr
& = \frac{1}{5}\ln \left| u \right| + C \cr
& {\text{back - substitute }}u = 1 + {x^5} \cr
& = \frac{1}{5}\ln \left| {1 + {x^5}} \right| + C \cr} $$