bIntegration by substitution/b is the reverse of the chain rule.
ul
liFind an appropriate expression for the substitution if the question does not suggest one./li
li(u = f(x) \implies \dfrac{du}{dx} = f'(x) )/li
liMake (dx) the subject: (dx = \dfrac{du}{f'(x)} )/li
liSubstitute (u) and (\dfrac{du}{f'(x)}) into the integration./li
liCancel any terms that contain (x)./li
liIf there are remaining terms that contain (x), rearrange (u = f(x)) and make (x) the subject, and make a further substitution./li
liWhen there are only terms containing (u) remaining, integrate normally./li
liFor indefinite integrals, unsubstitute (u=f(x)) to get the final answer./li
liFor 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. [/li]/ul
[b]uStandard results[/u]/b

(\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} )

[b]uIntegration by substitution standard results[/u]/b

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