Integration by substitution is the reverse of the chain rule.

• Find an appropriate expression for the substitution if the question does not suggest one.


• \(u = f(x) \implies \dfrac{du}{dx} = f'(x) \)


• Make \(dx\) the subject: \(dx = \dfrac{du}{f'(x)} \)


• Substitute \(u\) and \(\dfrac{du}{f'(x)}\) into the integration.


• Cancel any terms that contain \(x\).


• If there are remaining terms that contain \(x\), rearrange \(u = f(x)\) and make \(x\) the subject, and make a further substitution.


• When there are only terms containing \(u\) remaining, integrate normally.


• For indefinite integrals, unsubstitute \(u=f(x)\) to get the final answer.


• For definite integrals, the limits (with respect to \(x\)) must also be substituted into \(u=f(x)\) to get new limits (with respect to \(u\)). There is no need to unsubstitute as the final answer is a value.

Standard results

\(\displaystyle\int{\dfrac{f'(x)}{f(x)}} dx, \quad u = f(x) \)

\(u = f(x) \implies \dfrac{du}{dx} = f'(x) \)

\(dx = \dfrac{du}{f'(x)} \)

Substitute into integration:

\(\displaystyle\int{\dfrac{f'(x)}{f(x)}} dx \implies \displaystyle\int{\dfrac{f'(x)}{u}} \dfrac{du}{f'(x)} \)

\(\displaystyle\int{\dfrac{\cancel{f'(x)}}{u}} \dfrac{du}{\cancel{f'(x)}} = \displaystyle\int{\dfrac{1}{u}} du = \ln{|u|} + c \)

Unsubstitute \(u = f(x) \)

\(\implies \boxed{\displaystyle\int{\dfrac{f'(x)}{f(x)}} dx \implies \ln{|f(x)|} + c} \)