Integration by substitution

In calculus, integration by substitution is a method of evaluating an antiderivative or a definite integral by applying a change of variables. It is the integral counterpart of the chain rule for differentiation. For a definite integral, it can be shown as follows:

[math]\displaystyle{ \int_{a}^{b} f(\varphi(x))\varphi'(x)dx = \int_{\varphi(a)}^{\varphi(b)} f(u)du }[/math]

Where [math]\displaystyle{ u = \varphi (x) }[/math] such that [math]\displaystyle{ du = \varphi '(x)dx }[/math].

Steps

1. Let a variable equal part of the integrand, so that its derivative will cancel with the other part of the integrand
2. Apply the substitution
3. Evaluate the integral in terms of the new variable

Definite integral example

Consider the integral

[math]\displaystyle{ \int_{1}^{3} \tan(3x)dx }[/math]

Let [math]\displaystyle{ u = 3x }[/math] to obtain [math]\displaystyle{ du = 3dx }[/math] and [math]\displaystyle{ dx = \frac{du}{3} }[/math]. In this case, the [math]\displaystyle{ x }[/math] variable is not present, so the 1/3 can be factored out of the integrand. Since the integral is now in terms of [math]\displaystyle{ u }[/math], the bounds of integration (1 and 3 in this case), must be plugged in to the substitution u=3x. So the new bounds of integration are 3 and 9 to obtain,

[math]\displaystyle{ \begin{align} &\frac{1}{3} \int_{3}^{9} \tan(u)du = \frac{1}{3} (\ln|\sec(9)|-\ln|\sec(3)|)\\ &\approx 0.02767 \end{align} }[/math]

The antiderivative of [math]\displaystyle{ \tan{(u)} }[/math] may also be found using integration by substitution and ends up being [math]\displaystyle{ \ln{|\sec(u)|} }[/math].