Integration by Parts: A Beginner's Worked Guide

Short answer: integration by parts reverses the product rule: ∫u dv = uv − ∫v du. Pick u as the part that gets simpler when differentiated (use the order LIATE: Logs, Inverse trig, Algebraic, Trig, Exponential), let dv be the rest, then apply the formula.

Where the formula comes from

The product rule says (uv)′ = u′v + uv′. Integrate both sides: uv = ∫v du + ∫u dv. Rearranging gives the formula. So integration by parts is just the product rule run backwards — which is why it suits products like x·e^x. Not sure if parts is the right method? See when to use integration by parts.

The four-box method

Write four boxes before you start: u, du, dv, v. Filling them in first prevents most errors.

Worked examples

Example 1: ∫x e^x dx

u = x, du = dx; dv = e^xdx, v = e^x.
∫x e^x dx = x e^x − ∫e^x dx = x e^x − e^x + C.

Example 2: ∫x cos x dx

u = x, du = dx; dv = cos x dx, v = sin x.
= x sin x − ∫sin x dx = x sin x + cos x + C.

Example 3: ∫ln x dx (the hidden product)

Write it as ∫ln x · 1 dx. u = ln x, du = (1/x)dx; dv = dx, v = x.
= x ln x − ∫x · (1/x) dx = x ln x − x + C.

Example 4: ∫x² e^x dx (parts twice)

u = x², dv = e^xdx → x²e^x − ∫2x e^x dx. Apply Example 1 to the remaining integral:
= x²e^x − 2(x e^x − e^x) + C = e^x(x² − 2x + 2) + C.

Example 5: ∫e^x sin x dx (the loop)

Call the integral I. Parts with u = sin x: I = e^x sin x − ∫e^x cos x dx. Parts again with u = cos x: ∫e^xcos x dx = e^x cos x + I.
So I = e^x sin x − e^x cos x − I, giving I = ½e^x(sin x − cos x) + C.

Example 6: definite integral ∫₀¹ x e^2x dx

u = x, v = ½e^2x: [½x e^2x]₀¹ − ∫₀¹ ½e^2x dx = ½e² − [¼e^2x]₀¹ = ½e² − ¼e² + ¼ = (e² + 1)/4.

Beginner mistakes

• Choosing u = e^x in ∫x e^x dx — the new integral gets harder, not easier.
• Sign errors in the minus before ∫v du, especially when v is negative (like −cos x).
• Forgetting to evaluate uv at the limits in definite integrals.
• Dropping + C on indefinite integrals.

Practice section

1. ∫x sin x dx
2. ∫x e^3x dx
3. ∫x ln x dx
4. ∫arctan x dx

Now try the 8-question quiz. For the bigger picture see the integrals topic page, and use the free diagnostic to see whether integration is really your weak spot.