The video below works out some integration by parts
examples.
After watching the video, see if you can compute the same examples
without looking back at the video:
1) DO: Compute $\displaystyle \int x \ln(x)\,dx$,
letting $u=\ln(x)$ and $dv=x\,dx$. Be careful and precise with
where you write $u$ and $dv$ as in the video - do it the same way
every time to keep from making errors. (Notice that you cannot
let $dv=\ln x$, since you do not (yet) know the antiderivative of
$\ln x$.)
2) DO: Compute$\int
x \sin(x)\,dx$, letting $u=x$ and $dv=\sin(x)\, dx$.
3) DO: Set up $\int x \sin(x)\,dx$ (the same
integral as above), but this time letting $u=\sin x$ and $dv=x\,
dx$. What do you think about this integral $\displaystyle\int
v\,du$? It is fine to try one way,
then decide it might be better another way!
4) DO: Compute $\int x^2 e^x\,dx$. What
would you choose for $u$ and $dv$ and why? Remember, you
want your resulting integral $\displaystyle \int v\,du$ to be
simple to compute.
If you need help on these examples, rewatch the video.