Integration by Parts

The product rule in reverse

Integration by parts is derived from the product rule for derivatives. It's essential when you have a product of functions where one gets simpler when differentiated.

The IBP Formula

∫u dv = uv - ∫v du

Choose u (to differentiate) and dv (to integrate), then apply the formula.

Classic Example

Medium

Evaluate ∫x·eˣ dx

1

Choose u and dv

Let u = x (algebraic) → du = dx Let dv = eˣ dx → v = eˣ

2

Apply formula

∫x·eˣ dx = x·eˣ - ∫eˣ dx = x·eˣ - eˣ + C

3

Factor

= eˣ(x - 1) + C

Answer: eˣ(x - 1) + C

Key Takeaways

IBP: ∫u dv = uv - ∫v du
Use LIATE to choose u
Sometimes you need to apply IBP twice
Works great for x·eˣ, x·sin(x), ln(x), etc.