Integration by Parts

Reversing the product rule: ∫u dv = uv − ∫v du. The standard technique for integrating products of functions.

Integration by parts is the product rule run backward

d(uv) = u dv + v du

rearrange

∫ u dv = uv - ∫ v du

Trade the hard part

Choose u so differentiating makes it simpler, and choose dv so integrating is manageable.

Definition

Integration by Parts is the integral analogue of the product rule. For differentiable functions $u$ and $v$ :

$\int u\, dv = uv - \int v\, du$

How to use it:

LIATE rule for choosing $u$ (pick whichever comes first):

∫x·eˣ dx

Let $u = x$ , $dv = e^x\, dx$ . Then $du = dx$ , $v = e^x$ .

$\int x e^x\, dx = x e^x - \int e^x\, dx = x e^x - e^x + C = e^x(x - 1) + C$

Check: $\frac{d}{dx}[e^x(x-1)] = e^x(x-1) + e^x = xe^x$ ✓

Try it

Evaluate $\int x\cos x\, dx$ .

Solution

Let $u = x$ , $dv = \cos x\, dx$ . Then $du = dx$ , $v = \sin x$ .

$\int x\cos x\, dx = x\sin x - \int \sin x\, dx = x\sin x + \cos x + C$

Related concepts

Needs first