## Integral Calculus, Integration by Parts

### Integration by Parts

Another technique is integration by parts.
Let u(x) and v(x) be differentiable on a certain interval.
We would like to integrate the function u×v′,
but we don't know how.
By the
product rule,
the derivative of uv is uv′+vu′.
Thus the integral of uv′ + the integral of vu′ yields uv.
Write it this way.
∫ uv′ =
uv - ∫ vu′

We may know how to derive the latter integral, thus giving the former.

Assume we want to integrate log(x).
(Log is defined in the next section; its derivative is 1/x.)
Let u = log(x) and let v = x.
Thus log(x) is uv′.
By parts, the integral becomes x×log(x) - the integral of (x times the derivative of log(x)).
The second integrand reduces to 1,
hence the answer is x×log(x)-x.
Take the derivative to reproduce log(x).