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:

#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.