Integrating e^x and its friends

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

$$\boxed{\int e^x \ dx = e^x + C}$$

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

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

$$\int e^{3x} \ dx = \frac{e^{3x}}{3} + C.$$

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

More generally, it's worth remembering that:

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

We could also use substitution or eyeball to see that

$$\int e^{x^4}x^3 \ dx = \frac{e^{x^4}}{4} + C.$$