How do you test the improper integral intx^2 dx 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:
is divergent.
Explanation:
We have for
So:
so the improper integral is divergent.