Hi everyone! Today we are asked in class to solve an integral in anyway that we know. The given integral is $\displaystyle \int \frac{x^4}{x-1}dx$. She hints that it is possible to solve it using integration by parts, and I see that I might have to use tablature method to find the integral since I have to reduce $\displaystyle x^4$ to zero power. But I said, screw it! So I attempted to do it by doing somewhat partial fraction. So here's what I did:

1)I did a long division so that:

$\displaystyle \frac{x^4}{x-1}=x^3+x^2+x+1+\frac{1}{x-1}$

2)Turns out I don't have to solve for a partial fraction so:

$\displaystyle \int \frac{x^4}{x-1}dx= \int x^3+x^2+x+1+\frac{1}{x-1}dx$

3)So that if we integrate it, it becomes:

$\displaystyle \int \frac{x^4}{x-1}dx=\frac{x^4}{4}+\frac{x^3}{3}+\frac{x^2}{2}+x+l n\left |x-1\right |+c$

Is that solution good enough? Not really sure about it. Anyone got any ideas to evaluate the integral more efficiently? Thanks a lot in advance!