Integration by parts
The integral of u dv equals uv minus the integral of v du. It is the product rule run backwards, used when the integrand is a product of two different kinds of function.
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Analogy
Unpacking a parcel in two stages: you take out the big item, then deal with what is left.
Example
u = x, dv = e^x dx.du = dx, v = e^x.x e^x - integral of e^x dx.Result: x e^x - e^x + C.
In practice
Handles integrals such as x e^x or x sin x that substitution cannot touch.
Practice
Flashcards
- State the integration by parts formula.The integral of u dv = uv - the integral of v du.
- What is the integral of x e^x dx?x e^x - e^x + C.
Interview questions
- Evaluate the integral of x e^x dx and explain how you chose u and dv.
- Let u = x so du = dx
- Let dv = e^x dx so v = e^x
- Formula gives x e^x minus integral of e^x
- Answer x e^x - e^x + C
- Differentiate to check
Questions people ask
How is integration by parts related to the product rule?
Differentiating u times v gives u prime v plus u v prime. Integrate both sides and rearrange, and you get the by-parts formula. So it is the product rule in reverse, trading one integral for another that should be easier to do.
How do I decide what to call u?
Pick u as the part that gets simpler when differentiated, such as x or a logarithm. Pick dv as the part you can integrate easily. A common order for u is logarithms, then algebraic terms like x, then trig, then exponentials last.