Question #5f011

1 Answer
Oct 11, 2016

The integral does not converge for those limits.

Explanation:

int_(-oo)^oo(t^2dt)/(1 + t^2)t2dt1+t2 does not converge. It is easy to establish that. Note that t^2/(1 + t^2) gt 0t21+t2>0 for t ne 0t0 (it is an even function) and mainly, lim_(t->oo)t^2/(1 + t^2)=1

There is a t_0 such that for t > t_0->t^2/(1+t^2) > 1/2 for instance t_0 = 1 so considering T > t_0

int_(-T)^T(t^2dt)/(1 + t^2) = 2int_0^1(t^2dt)/(1 + t^2)+2int_1^T(t^2dt)/(1 + t^2) ge C_0+2/2(T-1)

with

C_0=2int_0^1(t^2dt)/(1 + t^2).

Finally

lim_(T->oo)int_(-T)^T(t^2dt)/(1 + t^2) ge C_0-lim_(T->oo)(T-1) = oo