# The Unapologetic Mathematician

## Indefinite Integration

Since we’ve established the connection between integration and antidifferentiation, we’ll be concerned mostly with antiderivatives more directly than derivatives. So, it’s useful to have some simple notation for antiderivatives.

That’s pretty much what the “indefinite integral” amounts to. It looks like an integral, and it does (what the FToC tells us is) all the hard work of integration, but it stops short of actually calculating an integral. Given a function $f(x)$, we write an antiderivative as $\int f(x)dx$. Note that we aren’t saying which antiderivative we mean, and for the purposes of the FToC (part 2), we don’t need to be. It’s customary, though, to write the result generically by adding a $+C$ to the end of it.

We know, for example, that

$\displaystyle\frac{d}{dx}\frac{x^{n+1}}{n+1}=\frac{(n+1)x^n}{n+1}=x^n$

Then we turn this around to write

$\displaystyle\int x^ndx=\frac{x^{n+1}}{n+1}+C$

and so on.

We can also go back and rewrite the two rules of integration we found before:

$\displaystyle\int f(x)+g(x)dx=\int f(x)dx+\int g(x)dx$
$\displaystyle\int cf(x)dx=c\int f(x)dx$

Notice here that we don’t need to add the $+C$, since each side consists of indefinite integrals. We can hide these “constants of integration” on both sides. They only need to show up once we fully evaluate an indefinite integral.