What is the Integral of (x+1)/x dx?

Jun 4, 2015

Not many people realize this, but the key to doing this is to separate it using the additive properties of integrals.

$\int \frac{x + 1}{x} \mathrm{dx}$

$= \int \cancel{\frac{x}{x}} \mathrm{dx} + \int \frac{1}{x} \mathrm{dx}$

$= \int \mathrm{dx} + \int \frac{1}{x} \mathrm{dx}$

$= \textcolor{b l u e}{x + \ln | x | + C}$

where you'll have to remember that the integral of $\frac{1}{u} \left(\frac{\mathrm{du}}{\mathrm{dx}}\right)$ is $\ln | u |$. You'll see it a lot more later.