Integration by Substitution

Many integrands are not standard forms as they stand, but become one after a change of variable. Integration by substitution replaces a chosen expression inside the integrand by a new variable $t$, so that the integral collapses to a standard integral in $t$.

The method

If the integrand can be written as $f\big(g(x)\big)\,g'(x)$, substitute

so that

After integrating in $t$, substitute $t=g(x)$ back to return to the original variable. The key skill is spotting that the derivative of the "inside" function $g(x)$ is already present (up to a constant) as a factor in the integrand.

A very useful special case: $\displaystyle\int \frac{f'(x)}{f(x)}\,dx$

When the numerator of a fraction is exactly the derivative of its denominator, the substitution $t=f(x)$ gives $dt=f'(x)\,dx$ and

So the integral of (derivative of denominator) over (denominator) is the natural logarithm of the denominator. Recognising this pattern turns many awkward-looking fractions into a one-line answer.