# What is int_3^oo 3/x -2/(x-2)dx?

Jun 12, 2017

Well, think about what this integral is asking you to do. You are starting at the left at $x = 3$ and integrating rightwards towards infinity... and the function decreases towards a horizontal asymptote...

If you integrate $\frac{3}{x} - \frac{2}{x - 2}$ from $3$ to $\infty$, you've attempted to integrate in a half-open interval, so the integral is not finite.

The graph looks like this:

graph{3/x - 2/(x-2) [-9.19, 19.29, -3.32, 10.91]}

$\int \frac{3}{x} - \frac{2}{x - 2} \mathrm{dx} = 3 \ln | x | - 2 \ln | x - 2 |$

Evaluating from $3$ to $\infty$ gives:

$= {\lim}_{x \to \infty} \left[3 \ln | x | - 2 \ln | x - 2 |\right] - \left[3 \ln | 3 | - {\cancel{2 \ln | 3 - 2 |}}^{0}\right]$

$= \infty - \infty ' - 3 \ln | 3 |$

$\implies$ $\textcolor{b l u e}{\text{DNE}}$