Integration by Parts

When the integrand is a product of two functions, none of the standard formulas or substitution usually applies directly — this is where integration by parts comes in, built by reversing the product rule of differentiation.

If $u$ and $v$ are differentiable functions of $x$, the product rule gives $\dfrac{d}{dx}(uv) = u\dfrac{dv}{dx} + v\dfrac{du}{dx}$. Integrating both sides and rearranging leads to the integration-by-parts formula. Writing the integrand as a product $f(x)\,g(x)$, with $f$ treated as the first function and $g$ as the second:

In words: the integral of a product equals (first function) times (integral of the second function), minus the integral of (derivative of the first function) times (integral of the second function). …