Integration by Parts

Reversing the Product Rule

Integration by parts reverses the product rule to handle integrals of products that substitution cannot touch. The formula is: $$\int u\, dv = uv - \int v\, du$$ Here, $u$ is differentiated and $dv$ is integrated, trading your original integral for a hopefully simpler one.

The art is choosing $u$ and $dv$ wisely. Use the mnemonic LIATE to order your choices for $u$:

Pick $u$ as the type appearing earliest in LIATE, and let $dv$ contain the rest.

Worked Example & Pitfalls

For $\int x e^x\, dx$, choose $u = x$ (algebraic, giving $du = dx$) and $dv = e^x\, dx$ (giving $v = e^x$). Applying the formula: $$\int x e^x\, dx = x e^x - \int e^x\, dx = x e^x - e^x + C$$ Sometimes parts must be applied twice, like for $\int x^2 e^x\, dx$.

Common pitfall: choosing $u$ and $dv$ badly, making $\int v\, du$ harder than the original. Always pick $u$ to get simpler when differentiated.

Recall the core split:

• Substitution is for compositions.
• Parts is for products.