Does $\int_{-\infty}^{\infty} \sin(t) \,dt $ converge? Does  $\int_{-\infty}^\infty \sin(t) \,dt $ converge or diverge? How would I prove it? 
Should I use 'principle value' to do:
$$\lim_{a \to \infty} \int_{-a}^a \sin(t)\,dt$$

$$\int_{-a}^a \sin(t)\,dt=0$$
for $a>0$ since sine is an odd function. Hence
$$\lim_{a \to \infty} \int_{-a}^a \sin(t)\,dt=0.$$
