# [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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