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Integration by PartsAfter $u$-substitution, integration by parts is 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 $u$ and $dv$ so that $v\,du$ is easier to integrate than $u\,dv$. While there aren't really hard-and-fast rules, the following hints help.
Remember, you can try various $u$ and $dv$, then differentiate
and antidifferentiate. If your second integral looks worse,
try again. Sometimes, we may have to integrate by parts
twice to get our answer. Example: Compute
$\displaystyle\int\frac{\ln(x)}{x^3}\,dx. $ The two most important things to
remember about integration by parts are and
2) do not be afraid of trial and
error. Once you decide to try parts,
over-thinking what to pick for $u$ and $dv$ takes longer than just
trying something. If your trial doesn't work, mark through
it and try something else. (Don't erase, or you'll forget
what you've tried!) The more you practice, the better feel
you'll get for how to choose. |