Integration by Substitution

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}

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