Q.Evaluate $\displaystyle\int x(x-3)^{4}\,dx$ using integration by parts.

Both $x$ and $(x-3)^{4}$ are algebraic, so by parts applies with $u=x$ chosen for its simple derivative.

Choose $u=x$ (the simpler factor) and $dv=(x-3)^{4}dx$:

Apply $\int u\,dv=uv-\int v\,du$:

Evaluate the remaining integral using formula 1: …

Check by differentiation: $\dfrac{d}{dx}\left[\dfrac{x(x-3)^{5}}{5}\right]=\dfrac{(x-3)^{5}}{5}+\dfrac{x\cdot5(x-3)^{4}}{5}=\dfrac{(x-3)^{5}}{5}+x(x-3)^{4}$, and $\dfrac{d}{dx}\left[-\dfrac{(x-3)^{6}}{30}\right]=-\dfrac{6(x-3)^{5}}{30}=-\dfrac{(x-3)^{5}}{5}$. Adding …

Choosing $u=(x-3)^{4}$ instead of $u=x$, which makes the by-parts process longer without simplifying anything (differentiating $(x-3)^{4}$ does not reduce it to a constant the way differentiating $x$ does). Also, dropping the $x-3$ shift and int …