∑Richard Hummel Math

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:

Choose $u$ and $dv$ from your integrand.
Compute $du$ (differentiate $u$ ) and $v$ (integrate $dv$ ).
Apply the formula: $\int u\, dv = uv - \int v\, du$ .
Hope that $\int v\, du$ is easier than the original integral.

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

Logarithms
Inverse trig
Algebraic (polynomials)
Trig
Exponentials

∫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

Calculus· Integration

IntegralsThe integral as accumulated area — Riemann sums, definite integrals, and the antiderivative as an operation that reverses differentiation.

Calculus· Integration

Fundamental Theorem of CalculusThe bridge connecting differentiation and integration — why antiderivatives compute areas and why the two operations are inverses of each other.

Calculus· Integration

Applications of IntegrationComputing areas between curves, volumes of solids of revolution, arc lengths, and other accumulated quantities.

Related reading

Calculus· Integration

The Fundamental Theorem, ExplainedWhy do antiderivatives compute areas? The Fundamental Theorem of Calculus connects two seemingly unrelated ideas, and understanding why it works changes how you see both.

Calculus· Integration

Integration as AccumulationThe integral is not just an area — it is a way of adding up infinitely many infinitely thin slices of anything. Distance, population, charge, probability: all of them can be computed by integration.