Integration by Parts

Some products cannot be integrated by any substitution because neither factor is (a constant multiple of) the derivative of the expression inside the other — a plain polynomial-times-exponential product such as $x\,e^{x}$ is the standard example. Integration by parts handles exactly this class of product.

The formula

If $u$ and $v$ are both functions of $x$, the product rule for differentiation, $\dfrac{d}{dx}(uv) = u\dfrac{dv}{dx}+v\dfrac{du}{dx}$, rearranges (after integrating both sides) into the integration-by-parts formula:

Choosing $u$ and $dv$

The usual guide for this chapter's problems is to let $u$ be the factor that gets SIMPLER when differentiated (a plain polynomial such as $x$, whose derivative is just $1$), and let $dv$ be the remaining factor, which must itself be integrable using the standard formulas already covered. Choosing the two the wrong way round makes the resulting integral harder, not easier.

Worked pattern: $\displaystyle\int x\,e^{x}\,dx$

Let $u=x$ and $dv=e^{x}\,dx$. Then $du=dx$ and $v=e^{x}$. Substituting into the formula:

Check by differentiating back: using the product rule, $\dfrac{d}{dx}\big[e^{x}(x-1)\big] = e^{x}(x-1) + e^{x}(1) = e^{x}(x-1+1) = x\,e^{x}$ — exactly the original integrand.