Chapter 7 - Techniques of Integration - 7.4 Integration of Rational Functions by Partial Fractions - 7.4 Exercises - Page 541: 7

\[\frac{1}{4}x^4+\frac{1}{3}x^3+\frac{1}{2}x^2+x+\ln|x-1|+C\]

Let \[I=\int\frac{x^4}{x-1}dx\] \[I=\int\frac{(x-1)(x^3+x^2+x+1)+1}{x-1}dx\] \[I=\int\left[x^3+x^2+x+1+\frac{1}{x-1}\right]dx\] \[I=\frac{1}{4}x^4+\frac{1}{3}x^3+\frac{1}{2}x^2+x+\ln|x-1|+C\] Hence \[I=\frac{1}{4}x^4+\frac{1}{3}x^3+\frac{1}{2}x^2+x+\ln|x-1|+C\].