Integration by parts
∫ u dv = uv − ∫ v du. Reverse of the product rule.
Y13. Use when integrand is a product of two functions, especially when one becomes simpler after differentiating (polynomial × exponential, polynomial × trig).
Worked examples
∫ x ex dx: u = x, dv = exdx; du = dx, v = ex; result = xex − ex + c.
∫ x sin x dx = −x cos x + sin x + c.
∫ ln x dx = x ln x − x + c. (u = ln x, dv = dx)
Frequently asked questions
How to choose u and dv?
‘LIATE’ rule: pick u in this priority order: Logarithmic, Inverse trig, Algebraic, Trig, Exponential.
When to apply twice?
Polynomial × trig or exp may need parts twice. e.g. ∫ x² ex dx.