Question

How do you evaluate the integral `int x dx` from `-oo` to `oo` if it converges?

Answer 1

`0`

Explanation:

We have:

`int_(-oo)^ooxdx`

Integrating:

`=[x^2/2]_(-oo)^oo`

We can't technically "plug in" infinity and negative infinity, so take their limits:

`=lim_(xrarroo)(x^2/2)-lim_(xrarr-oo)(x^2/2)`

Both approach positive infinity:

`=oo-oo=0`

This should make sense, if we think about `int_(-oo)^ooxdx` as describing the area between the line `y=x` and the `x`-axis. The negative portion where `x<0` is completely balanced out by the positive area where `x>0`:

graph{x [-304.4, 304.4, -152.2, 152.2]}