Integration by parts: undo the product rule

Integration by parts is for a product of two different kinds of function, where no simpler method works:

• $x\sin 2x$: a polynomial times a trig function;
• $x^2e^{-x}$: a polynomial times an exponential;
• $x\tan^{-1}x$: a polynomial times inverse tan;
• $\ln x$: this one's a product in disguise, $1 \times \ln x$.

The idea: differentiate one factor (to make it simpler) and integrate the other.

Pick $u$ so that differentiating it makes things simpler. In order of preference:

1. $\ln x$, if there is one. (You can't easily integrate it, but its derivative $\tfrac1x$ is simple.)
2. $\tan^{-1}x$, if there is one. (Same reason: its derivative is $\tfrac{1}{1 + x^2}$.)
3. Otherwise, the polynomial part, like $x$ or $x^2$. Differentiating lowers its power.

Everything else is $\dfrac{dv}{dx}$.

1. Write down $u$ and $\dfrac{dv}{dx}$.
2. Work out $\dfrac{du}{dx}$ (differentiate $u$) and $v$ (integrate $\dfrac{dv}{dx}$, no $+\,C$ yet).
3. Put them into the formula: $uv - \displaystyle\int v\,\frac{du}{dx}\,dx$.
4. Do the new integral. If it's still a product with a polynomial, use parts again.
5. Add $+\,C$ at the end, or put in the limits.