# What is the antiderivative of 2/x?

Mar 26, 2016

$2 \ln \left\mid x \right\mid + C$

#### Explanation:

Begin by writing the problem in mathspeak:
$\int \frac{2}{x} \mathrm{dx}$

We know that $\int \frac{1}{x} \mathrm{dx} = \ln \left\mid x \right\mid + C$; so how can we simplify $\int \frac{2}{x} \mathrm{dx}$ to $\int \frac{1}{x} \mathrm{dx}$? Well, there's a rule for integrals that says: $\int c x \mathrm{dx} = c \int x \mathrm{dx}$; in other words, we can pull constants out of the integral. Because $2$ is a constant, we can bring it out:
$= 2 \int \frac{1}{x} \mathrm{dx}$

And applying the antiderivative of $\frac{1}{x}$, we have:
$2 \left(\ln \left\mid x \right\mid + C\right) = 2 \ln \left\mid x \right\mid + C \to$(note that $2 C$ is just another constant, so we can write it as $C$)