How do you test the improper integral #int x^3 dx# from #(-oo, oo)# and evaluate if possible?

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Answer 1 · 2017-08-02T23:37:45

The integral is not convergent as:

#int_(-oo)^(+oo) x^3 dx = lim_(u->oo) int_(-u)^0 x^3dx + lim_(v->oo) int_0^v x^3dx#

#int_(-oo)^(+oo) x^3 dx = lim_(u->oo) -u^4/4 + lim_(v->oo) v^4/4#

The two limits should be finite separately and they are not.

The integral is however convergent in the sense of Cauchy's principal values as #x^3# is an odd function, so:

#int_(-t)^t x^3dx = [x^4/4]_(-t)^t = 0#

and then:

#lim_(t->oo) int_(-t)^t x^3dx = 0#