Answer
Improper.
(Improper Integrals with Infinite Integration Limits, case 3)
Work Step by Step
Looking at the Definition of Improper Integrals with Infinite Integration Limits,
and comparing the given integral with
3. If $f$ is continuous on the interval $(-\infty, \infty)$, then
$\displaystyle \int_{-\infty}^{\infty}f(x)dx=\int_{-\infty}^{c}f(x)dx+\int_{c}^{\infty} f(x)dx$, where $c$ is any real number
we find that
$f(x)= \displaystyle \frac{\sin x}{4+x^{2}}$ is continuous on the interval $(-\infty, \infty)$
because the denominator is never zero (it is at least 4),
and, the bounds of the integral are $\pm\infty.$
So, this is an improper integral.
