# How do you find the indefinite integral of int 10/x?

Feb 26, 2018

$10 \int \frac{\mathrm{dx}}{x} = 10 \ln | x | + C$

#### Explanation:

Factor $10$ outside of the integral; this can be done as $10$ is just a constant. Doing this cleans up the work a little, although it doesn't change the final answer.

$10 \int \frac{\mathrm{dx}}{x}$

Recall that $\int \frac{\mathrm{dx}}{x} = \ln | x | + C$.

First, note that we have the absolute value sign around $x$ because $\ln \left(x\right)$ doesn't exist if $x$ is negative. We want to avoid that. Furthermore, note that the natural log function is appropriate, because differentiating $\ln \left(x\right)$ gives us back $\frac{1}{x} ,$ the function we were originally trying to integrate. Finally, don't forget the constant of integration, $C$, as we must account for any possible constants (there are infinitely many -- the derivative of a constant is always $0$, $C$ could be any value).

$10 \int \frac{\mathrm{dx}}{x} = 10 \ln | x | + C$

Feb 26, 2018

$\int \frac{10}{x} \mathrm{dx} = 10 \ln \left(\left\mid x \right\mid\right) + C$

#### Explanation:

$\int \frac{10}{x} \mathrm{dx}$ can be converted and made simpler to $10 \int \frac{1}{x} \mathrm{dx}$

$\int \frac{1}{x} \mathrm{dx}$ is a basic integral, which equals $\ln \left(\left\mid x \right\mid\right)$

therefore, $\int \frac{10}{x} \mathrm{dx} = 10 \ln \left(\left\mid x \right\mid\right) + C$