How do you test the improper integral intx^2 dxx2dx from (-oo, oo)(,) and evaluate if possible?

2 Answers
Apr 11, 2017

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)

Apr 11, 2017

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.