Integration by Substitution

Not every integral can be evaluated directly from the standard formulas — many integrands are composite functions in disguise, and integration by substitution is the technique for simplifying them by changing the variable of integration.

The idea: if we substitute $g(x) = t$, then differentiating gives $g'(x)\,dx = dt$. This lets us rewrite an integral of the form $\int f(g(x))\,g'(x)\,dx$ entirely in terms of $t$:

Once the integral is evaluated in terms of $t$, we substitute $t = g(x)$ back to express the answer in terms of $x$. For instance, in $\int \dfrac{2x}{x^2+1}\,dx$, putting $t = x^2+1$ gives $dt = 2x\,dx$, turning the integral into the elementary $\int \dfrac{1}{t}\,dt$.

Recognising which substitution to try is mostly a matter of pattern recognition. A few substitutions that work well in common situations:

• For $\sqrt{f(x)}$, put $f(x) = t$ (or $t^2$).
• For $\log x$, put $\log x = t$, or equivalently $x = e^t$.
• For a composite $f(g(x))$, put $g(x) = t$.
• For $[f(x)]^{m/n}$, put $f(x) = t^n$. …