Integration by parts 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:

• Assign one function to \(u\) and the other function \(\dfrac{dv}{dx}\).


• The function which has a "simpler" derivative should be assigned to \(u\).


• Differentiate \(u\) to get \(\dfrac{du}{dx}\) and integrate \(\dfrac{dv}{dx}\) with respect to \(x\) to get \(v\).


• Substitute \(u\), \(v\), \(\dfrac{du}{dx}\) and \(\dfrac{dv}{dx}\) into the parts formula.


• Evaluate \(\displaystyle\int{v\dfrac{du}{dx}} dx \).


• If this is impossible to do, try switching the functions assigned to \(u\) and \(\dfrac{dv}{dx} \).


• You may have to apply the parts formula again to integrate it.