Note 6.1 - Integration by Parts
1 Introduction
Integration by parts is essentially the inverse of product rule. It turns an inte- gration into another hopefully easier one.
2 The Formula
Start with the product rule for differentiation d
dxuv = udv dx+ vdu
dx,
3 Instruction Manual and Principles
The instructions are obvious. When we see an expression f (x)dx that can not be integrated right away, name some part of f to be u and the rest dv. Apply the formula above and see if we get an easier integral. Here is a quick example:
1
However, there are often more than one way to name u and dv. Some of them are not helpful:
So we usually prefer to set u to be something that becomes nicer after differentiation (e.g. polynomials).
4 Examples
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