###### Example 10.3.1

Use the method of substitution to find \(\displaystyle \int e^{3x} \, dx\text{.}\)

The integrand \(e^{3x}\) involves composition with \(u=3x\text{.}\) This is our substitution variable. Because \(u'=3\text{,}\) we have \(du = 3 dx\) so that \(\displaystyle dx = \frac{du}{3}\text{.}\) We rewrite the integral in terms of the substitution variable \(u\text{.}\) After antidifferentiation using the variable \(u\text{,}\) we back-substitute our original formula for \(u=3x\text{.}\) The work is shown below.