Integration by Parts
Simple Explanation
When an integrand is a product of two different kinds of functions (like a polynomial times an exponential), integration by parts — ∫u dv = uv − ∫v du — trades the original hard integral for a hopefully-easier one.
Why Do We Need It?
Substitution alone cannot handle every product integral — integration by parts covers a wide range of cases substitution cannot, particularly products involving polynomials, exponentials, logarithms, and trigonometric functions.
Formula
Integration by Parts
∫u dv = uv − ∫v du
Reverses the Product Rule for differentiation, turning one hard integral (∫u dv) into a simpler one (∫v du) plus an easy term (uv).
- u
- — the factor chosen to be differentiated (should get simpler when differentiated)
- dv
- — the factor chosen to be integrated (should be easy to integrate)
When to use it: Whenever the integrand is a product of two different types of functions (e.g. a polynomial times an exponential, or a polynomial times a trigonometric function) that substitution cannot simplify.
Worked Example
Integrate a product using integration by parts
Find ∫x·eˣ dx.
Why Does This Work?
This is exactly the Product Rule for differentiation, rearranged and integrated: d/dx[uv] = u'v + uv', so integrating both sides gives uv = ∫v du + ∫u dv, which rearranges into the Integration by Parts formula.
Real-Life Example
Computing work done by a changing force
The work done by a force that varies with position, combined with a displacement that also depends on another changing quantity, often produces a product of two different function types to integrate.
Integration by parts is the standard tool physicists and engineers reach for whenever such a product integral cannot be simplified by substitution alone.
Practice
Find ∫x·cos(x) dx.
HardCommon mistake
Choosing u and dv the wrong way around — if u is chosen so that it gets more complicated (not simpler) when differentiated, the remaining integral ∫v du usually becomes harder rather than easier.
Quick Review
- ∫u dv = uv − ∫v du.
- Pick u to simplify under differentiation, and dv to be easy to integrate.
- Comes from reversing the Product Rule for differentiation.