# Integration by Substitution

April 27, 2020
Today's lesson was requested by Oscar Flores via YouTube.  You can watch the full response to Oscar below.
Oscar wanted to know how to integrate the following:
\begin{align} \int x^2e^{x^3}dx \end{align}
The first thing to recognise is the connection between the $x^3$ and the $x^2$.  When you differentiate an $x^3$ term you will get an $x^2$ term.  Experience completing these sorts of questions therefore tells me that a substitution can be used as follows:
\begin{align} \int x^2e^{x^3}dx& \\\\ \text{Let } u&=x^3\\\\ \frac{du}{dx}&=3x^2\\\\ dx&=\frac{du}{3x^2}\\\\\end{align}
We can now substitute in values as follows:
\begin{align} \int x^2e^{x^3}dx&=\int x^2e^u \frac{du}{3x^2} \\\\ &=\frac{1}{3}\int e^u du\\\\ &=\frac{e^u}{3}+c \end{align}
The last step is to substitute back in for $u$:
\begin{align} \int x^2e^{x^3}dx&=\frac{e^{x^3}}{3}+c \end{align}