bIntegration by parts/b is the reverse of the product rule.

Consider the product rule:

(y = uv \implies \dfrac{dy}{dx} = v\dfrac{du}{dx} + u\dfrac{dv}{dx} )

Integrate all terms with respect to (x):

(\displaystyle\int{\dfrac{dy}{dx}} dx = \displaystyle\int{v\dfrac{du}{dx}} dx + \displaystyle\int{u\dfrac{dv}{dx}} dx )

Rearrange and simplify:

(\displaystyle\int{u\dfrac{dv}{dx}} dx = y - \displaystyle\int{v\dfrac{du}{dx}} dx )

(\implies \boxed{\displaystyle\int{u\dfrac{dv}{dx}} dx = uv - \displaystyle\int{v\dfrac{du}{dx}} dx} )

To use this formula: ulliAssign one function to (u) and the other function (\dfrac{dv}{dx}). /liliThe function which has a "simpler" derivative should be assigned to (u). /liliDifferentiate (u) to get (\dfrac{du}{dx}) and integrate (\dfrac{dv}{dx}) with respect to (x) to get (v). /liliSubstitute (u), (v), (\dfrac{du}{dx}) and (\dfrac{dv}{dx}) into the parts formula./liliEvaluate (\displaystyle\int{v\dfrac{du}{dx}} dx )./liliIf this is impossible to do, try switching the functions assigned to (u) and (\dfrac{dv}{dx} )./liliYou may have to apply the parts formula again to integrate it./li/ul

[b]uIntegration by parts[/u]/b

(\displaystyle\int{u\dfrac{dv}{dx}} dx = uv - \displaystyle\int{v\dfrac{du}{dx}} dx )