Question

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

Answer 1

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 `3/x - 2/(x-2)` from `3` to `oo`, 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]}

and Wolfram Alpha gives...

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`int 3/x - 2/(x-2)dx = 3ln|x| - 2ln|x-2|`

Evaluating from `3` to `oo` gives:

`= lim_(x->oo) [3ln|x| - 2ln|x-2|] - [3ln|3| - cancel(2ln|3-2|)^(0)]`

`= oo - oo' - 3ln|3|`

`=>` `color(blue)("DNE")`