Question

What is `int_-oo^oo 3/x dx`?

Answer 1

One might think of splitting this into two integrals at first:

`int_(-oo)^(0) 3/xdx + int_(0)^(oo) 3/xdx`

This integral is fairly easy as an indefinite integral if you recall the antiderivative of `1/x`; `int 1/xdx = ln|x|`. The challenge is evaluating the boundaries, if it is possible.

`=> |[3ln|x|]|_(-oo)^(0) + |[3ln|x|]|_(0)^(oo)`

`= ([3ln|0|] - lim_(x->-oo) ln|x|) + (lim_(x->oo) ln|x| - [3ln|0|])`

`= cancel(3ln|0|) - lim_(x->-oo) ln|x| + lim_(x->oo) ln|x| - cancel(3ln|0|)`

`= -3lim_(x->-oo) ln|x| + 3lim_(x->oo)ln|x|`

but since it is `|x|`, `lim_(x->oo) = lim_(x->-oo)`:

`= 0*lim_(x->oo)ln|x|`

`= 0*oo`

`=>` does not converge