Integration by Parts — Mathematics Knowledge System

Overview

Integration by parts is the integral counterpart of the product rule, used to integrate a product of two functions when no simpler technique applies. It transforms an integral into a (hopefully simpler) different integral, by shifting the derivative from one factor onto the other.

Intuition

The product rule says (uv)' = u'v + uv'. Integrating both sides recovers uv on the left, so ∫u'v dx + ∫uv' dx = uv. Rearranged, this lets you trade the integral of uv' for the (hopefully easier) integral of u'v — you're essentially moving the derivative from one factor to the other in exchange for a sign-free swap.

Interactive Graph

Formal Definition

Definition

Derived directly from the product rule, integration by parts states:

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

Choose u and dv from the integrand so that ∫v du is easier than the original integral

Integration by Parts

Derivation

Starting from the product rule and integrating both sides:

\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)

Product rule

\int \frac{d}{dx}[uv]\,dx = \int u'v\,dx + \int uv'\,dx

Integrate both sides with respect to x

u v = \int v\,du + \int u\,dv

The left side integrates to uv; rewrite u'dx as du and v'dx as dv

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

Solve for the desired integral

Applications

Computing moments of inertia and center of mass for non-uniform density distributions often requires integrating products like x·f(x), resolved via integration by parts.

Worked Examples

Choose u=x (gets simpler when differentiated), dv=e^x dx. Then du=dx, v=e^x.
$u=x,\ dv=e^x\,dx \implies du=dx,\ v=e^x$
Apply the integration by parts formula.
$\int x e^x\,dx = xe^x - \int e^x\,dx$
Evaluate the remaining integral and add the constant.
$= xe^x - e^x + C$

Answer: xe^x - e^x + C

Practice Problems

Difficulty 4/10

Evaluate ∫x sin(x) dx.

Difficulty 6/10

Evaluate ∫x² e^x dx (requires integration by parts twice).

Common Mistakes

Common Mistake

Choosing u and dv poorly, leading to a more complicated integral than the original.

Use the LIATE heuristic (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) to prioritize which factor to set as u — generally pick u to be the type that simplifies upon differentiation.

Common Mistake

Forgetting the minus sign in front of ∫v du.

The formula is ∫u dv = uv − ∫v du; dropping the negative sign is one of the most common errors.

Quiz

Integration by parts is the integral form of which differentiation rule?

the chain rulethe product rulethe quotient rulethe power rule

A common guideline for choosing u in ∫u dv is 'LIATE'. What does picking u correctly aim to do?

Make the remaining integral ∫v du simpler than the originalMake u as large as possibleEliminate the constant of integrationAvoid using v

Summary

Integration by parts: ∫u dv = uv − ∫v du, derived from the product rule.
Choose u to be the factor that simplifies when differentiated (LIATE priority), and dv to be the remaining factor that's easy to integrate.
Some integrals (like x²eˣ) require applying the technique more than once.
∫ln(x)dx and similar single-function integrals can be integrated by parts using dv=dx.

References

WebsiteWikipedia — Integration by parts