Integration by Parts

Reversing the Product Rule

Integration by parts is the integration technique that reverses the product rule. It handles integrals of products — especially a polynomial times an exponential, logarithm, or trig function — that substitution cannot touch. The formula, derived by integrating the product rule, is: $$\int u\, dv = uv - \int v\, du.$$ You split the integrand into a part $u$ (to differentiate) and a part $dv$ (to integrate), then trade the original integral for a hopefully simpler one, $\int v\, du$.

The art is choosing $u$ and $dv$ wisely — you want $\int v\, du$ to be easier than what you started with. A useful guide is the mnemonic LIATE, ordering good choices for $u$: Logarithmic, Inverse trig, Algebraic (polynomial), Trigonometric, Exponential. Pick $u$ to be whichever type appears earliest in this list, and let $dv$ be the rest.

For example, $\int x\, e^x\, dx$: choose $u = x$ (algebraic, so $du = dx$) and $dv = e^x\, dx$ (so $v = e^x$). Then $$\int x e^x\, dx = x e^x - \int e^x\, dx = x e^x - e^x + C.$$ The new integral $\int e^x\, dx$ is trivial — a good sign the choice was right. Sometimes parts must be applied twice (e.g. $\int x^2 e^x\, dx$), and occasionally it produces the original integral again, which you solve algebraically.

Substitution and parts are the two workhorses: substitution for compositions, parts for products. Recognizing which structure an integral has is the first step in choosing the technique.

Common pitfall: choosing $u$ and $dv$ badly, so the new integral $\int v\, du$ is harder than the original. The goal is to make it simpler — pick $u$ to be the part that gets simpler when differentiated (a polynomial eventually becomes a constant), following LIATE. Choosing $u = e^x$ in $\int x e^x\, dx$, for instance, leads nowhere; the right choice ($u = x$) makes the remaining integral trivial.