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Integration by PartsAfter $u$substitution, integration by parts is by far the most important technique 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 $dv$. We want to choose $u$ so that $du$ is simpler than $u$, and $dv$ so that $v$ isn't much more complicated than $dv$. Remember that:
If we 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 we have two terms at the same level, then we can try using some algebra to relate $\int v\,du$ to $\int u\,dv$. We may have to integrate by parts twice to make this work.
