Show that the integral does not converge Show that $\int_0^\infty \sin(x) \,$ does not converge.
$$\int_0^k \sin(x) \leq \int_0^k \, dx = k$$
$$\lim_{k \to \infty} k = \infty$$
So this integral does not converge.
Is this correct?

big hint
$$\int_0^k\sin(x)\mathrm d x=1-\cos(k)$$
Then $\int_0^\infty \sin(x)\mathrm d x$ converge if and only if $\lim_{k\to \infty }\cos(k)$ exist. 
