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Integration by parts

Integration by parts is the product rule used backwards. You choose one factor to differentiate and one to integrate. Choose to differentiate the factor that becomes simpler, often xx or ln⁡x\ln x. After the formula there is a new integral, and it should be easier than the one you started with. A definite integral gets limits on both terms.

Try it yourself

The integral as area

∫u′v dx=uv−∫uv′ dx\int u'v\,dx = uv - \int uv'\,dxintegration by parts

Symbols

u′u'the factor you integrate
vvthe factor you differentiate

Example

∫1eln⁡x dx\int_1^e \ln x\,dx: with v=ln⁡xv = \ln x, [xln⁡x−x]1e=(e−e)−(0−1)=1[x\ln x - x]_1^e = (e - e) - (0 - 1) = 1.

If the new integral is harder, you chose the wrong factor to differentiate.
Practise integration for free →

← The definite integral · Arithmetic sequence and series →

Part of Mathematics R2: Integration.