Answer
The integral is improper.
(Improper Integrals with Infinite Integration Limits, case 1)
Work Step by Step
Looking at the Definition of Improper Integrals with Infinite Integration Limits,
and comparing with the given integral,
$f(x)=\cos x$ is continuous on the interval $[a, \infty)$, and the upper bound is $\infty$,
so case (1) applies,
$\displaystyle \int_{a}^{\infty}f(x)d\mathrm{x}=\lim_{b\rightarrow\infty}\int_{a}^{b}f(x)dx$.
