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Integrating ex and its friends

You're going to like this part. Since $\displaystyle \frac{d}{dx} (e^x) = e^x$, we know immediately that:

\begin{displaymath}\boxed{\int e^x \ dx = e^x + C}\end{displaymath}

In other words, if ex is its own mother, then it's its own child, too.

Now, we can find lots of other integrals using substitution. For instance,

\begin{displaymath}\int e^{3x} \ dx = \frac{e^{3x}}{3} + C.\end{displaymath}

(You know, u=3x, du=3dx, etc. Usual drill.)

More generally, it's worth remembering that:

\begin{displaymath}\int e^{kx} \ dx = \frac{e^{kx}}{k} + C \quad \text{ for any constant $k$ other than 0}.\end{displaymath}

We could also use substitution or eyeball to see that

\begin{displaymath}\int e^{x^4}x^3 \ dx = \frac{e^{x^4}}{4} + C.\end{displaymath}

Joel Hass