6.11 Integrating Using Integration by Parts

The Integration by Parts formula

Integration by parts reverses the product rule: ∫u dv = uv − ∫v du. Choose u and dv from the integrand, compute du (derivative of u) and v (antiderivative of dv). Use it when the integrand is a product of two different types of functions, like x·eˣ or x·sin(x), that basic rules can't handle.

Choosing u wisely: LIATE

Pick u using the LIATE guide (Logarithmic, Inverse trig, Algebraic, Trig, Exponential) — functions earlier in the list are usually better choices for u because they simplify when differentiated. Choosing poorly can make the new integral harder instead of easier, so check that ∫v du is simpler than the original.

Repeated and definite applications

Some integrals need repeated integration by parts (e.g., x²eˣ) until the remaining integral is solvable. For definite integrals, apply the formula as ∫ₐᵇ u dv = [uv]ₐᵇ − ∫ₐᵇ v du, evaluating the uv term at the bounds. Watch for cases where the process cycles back to the original integral, allowing algebraic solving.

Key terms

Integration by parts formula

∫u dv = uv − ∫v du, derived from the product rule for derivatives.

LIATE

A guide for choosing u: Logarithmic, Inverse trig, Algebraic, Trig, Exponential — earlier types are usually better choices for u.

When to use integration by parts

When the integrand is a product of two different function types (e.g., x·sin(x), x²·eˣ) not solvable by basic rules.

Definite integral by parts

∫ₐᵇ u dv = [uv]ₐᵇ − ∫ₐᵇ v du; evaluate the uv term at the bounds.

Repeated integration by parts

Applying the formula more than once when the resulting integral is still a product, such as with x²eˣ or x²sin(x).

Try an AP-style question

∫ x eˣ dx = ?

1. eˣ + C
2. xeˣ − eˣ + C
3. x²eˣ/2 + C
4. xeˣ + eˣ + C