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[TIL] Integration by parts



Integration by parts or partial integration is a theorem to help us solve complex Integration problem of a product of functions. If a function can represent as a product of a function \(u(x)\) and a derivative of a function \(u′(x)\) , we can use integration by parts.

If \(u = u(x)\) and \(du = u′(x) dx\), while \(v = v(x)\) and \(dv = v′(x) dx\), then integration by parts states that:

\(\int_a^b u(x) v'(x) \, dx\ = [u(x) v(x)]_a^b-\int_a^b v(x) u'(x) \, dx \).

or more compactly:

\(\int u\,dv=uv-\int v\,du\).

This method is very useful when sometimes integration of \(v(x)u′(x) \) is easier to find.



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