Choosing u and dv
A good choice makes the new integral simpler than the old one. Logarithms and inverse trig functions go in u. Exponentials and sines are usually dv.
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Analogy
Choosing which piece of flat-pack furniture to assemble first so the rest fits easily.
Example
u = ln x, dv = x dx.du = dx/x, v = x^2/2.(x^2/2) ln x - integral of x/2 dx.Result: (x^2/2) ln x - x^2/4 + C.
In practice
A wrong choice makes the integral harder, which tells you to swap the roles.
Practice
Flashcards
- Which function is usually u: ln x or x?ln x, because it simplifies when differentiated.
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
Why can't I integrate ln x directly by a table?
The natural log has no simple antiderivative from the standard list. But by parts works with u = ln x and dv = dx. This gives x ln x minus x, plus C. Check by differentiating: the result is ln x, as needed.
What if integrating by parts makes things worse?
Swap your choice of u and dv. If the new integral looks harder than the original, it was a poor choice. Sometimes you need to apply the method twice, as with x squared times e to the x, lowering the power of x each time.