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}

Add your question below or leave a comment. I will endeavour to respond.

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