Question

Question #5f011

Answer 1

The integral does not converge for those limits.

Explanation:

`int_(-oo)^oo(t^2dt)/(1 + t^2)` does not converge. It is easy to establish that. Note that `t^2/(1 + t^2) gt 0` for `t ne 0` (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`