Question

How do you test the improper integral `intx^2 dx` from `(-oo, oo)` and evaluate if possible?

Answer 1

It is divergent.

Explanation:

We deal with the indefinite integral as normal: So:

`int \ x^2 \ dx = 1/3x^3 \ \ (+c)`

Then as we are dealing with an infinite integration limit we use the limit definition to get:

`int_(-oo)^(oo) \ x^2 \ dx = [ \ 1/3x^3 \ ]_(-oo)^(oo)`

`" " = lim_(n rarr oo)[ \ 1/3x^3 \ ]_(-n)^(n)`

`" " = lim_(n rarr oo)1/3(n^3-(-n)^3)`

`" " = lim_(n rarr oo)1/3(n^3+n^3)`

`" " = lim_(n rarr oo)2/3n^3`

Which is clearly divergent (and therefore undefined)

Answer 2

The integral:

`int_(-oo)^oo x^2dx`

is divergent.

Explanation:

We have for `t > 0`:

`int_(-t)^t x^2dx = [x^3/3]_(-t)^t = t^3/3 +t^3/3 = 2/3 t^3`

So:

`int_(-oo)^oo x^2dx = lim_(t->oo) int_(-t)^t x^2dx = lim_(t->oo) 2/3 t^3 = +oo`

so the improper integral is divergent.