# Integration using substitution

When we know an integral

\begin{equation}
\int e^x \d{x} = e^x + c
\end{equation}

we can evaluate related integrals like:

\begin{equation}
\int e^{-x} \d{x} = -e^{-x} + c
\end{equation}

by making a *substitution*. Specifically, we define a new variable *u*,
calculate its differential, and then replace both. For example, in this case,
let $u = -x$. Then, $\d{u} = -\d{x}$. Equivalently, $x = -u$ and
$\d{x} = -\d{u}$ so

\begin{equation}
\int e^{-x} \d{x} = -\int e^u \d{u} = -e^u + c = -e^{-x} + c
\end{equation}

```{example} *u*-substitution
Evaluate the integral

\begin{equation}
\int x e^{x^2} \d{x}
\end{equation}

---

Let $u=x^2$ so $\d{u} = 2x \d{x}$. Then,

\begin{align}
\int e^{x^2} x\d{x} =  \int e^u \cdot \frac{1}{2}\d{u} = \frac{1}{2} e^u + c
\end{align}

Plugging *u* back in gives the final answer:
\begin{equation}
\int xe^{x^2} \d{x} = \frac{1}{2}e^{x^2} + c
\end{equation}
```