Integration by Substitution

Some integrands are not directly one of the standard forms, but become one after a change of variable. Integration by substitution (also called the change-of-variable method) is the technique for spotting and using this.

When substitution applies

Substitution works cleanly whenever the integrand can be seen as a function of some inner expression $g(x)$, multiplied by (a constant multiple of) $g'(x)$ — exactly what the chain rule produces when differentiating a composite function, run in reverse:

After integrating with respect to $u$ using the standard formulas, the last step is always to substitute $u=g(x)$ back, so the final answer is written in terms of the original variable $x$ — an integral is never left in terms of $u$.

Worked pattern: $\displaystyle\int 3x^{2}(x^{3}+4)^{4}\,dx$

Let $u = x^{3}+4$, so $du = 3x^{2}\,dx$ — and $3x^{2}\,dx$ is EXACTLY what multiplies $(x^{3}+4)^{4}$ in the integrand, so the substitution is a clean fit:

Check by differentiating back: using the chain rule, $\dfrac{d}{dx}\left[\dfrac{(x^{3}+4)^{5}}{5}\right] = \dfrac{5(x^{3}+4)^{4}\cdot 3x^{2}}{5} = 3x^{2}(x^{3}+4)^{4}$, exactly the original integrand.

Spotting the right substitution …