After $u$-substitution, integration by parts is by far the most important technique for you to learn. It converts a hard integral $\int u\,dv$ into an easier integral $uv - \int v\,du$. The tricky thing is figuring out what to pick for $u$ and $dv$. Pick $u$ so that $du$ is simpler than $u$, preferably a lot simpler, and pick $dv$ so that $v$ isn't much more complicated than $dv$. Remember that:

• Some functions get a lot simpler when differentiated or a lot more complicated when integrated. These almost always want to be part of $u$.
• Some functions get a little simpler when differentiated. These are a lower priority to include in $u$.
• Some functions, like $e^x$ and $\sin(x)$, don't get simpler at all. These have still lower priority, and usually are part of $dv$.
• Some functions actually get simpler when integrated and are almost always part of $dv$.

If you can pick $u$ from higher on the list and $dv$ from lower on the list, then $\int v\,du$ will be simpler than $\int u\,dv$.

If you have two terms at the same level, then try using algebra to relate $\int v\,du$ to $\int u\,dv$. You may have to integrate by parts twice to make this work.