Does $\int_{-\infty}^{\infty} \sin(t) \,dt $ converge?

Question

Does $\int_{-\infty}^\infty \sin(t) \,dt $ converge or diverge? How would I prove it?

Should I use 'principle value' to do: $$\lim_{a \to \infty} \int_{-a}^a \sin(t)\,dt$$

Answer 1

Does $\int_{-\infty}^\infty \sin (t)\,\mathrm{d}t$ converge?

No, for the limit to exist with limit $L$, then for every pair of sequences $a_n\to-\infty$ and $b_n\to\infty$, we would have that

$$\lim\limits_{n\to\infty} \int_{a_n}^{b_n} \sin (t)\,\mathrm{d}t = L$$ which is obviously not the case.

The Cauchy Principal Value is different to usual convergence and this value does exist, it is $$\lim\limits_{a\to\infty}\int_{-a}^a\sin (t) \,\mathrm{d} t = 0.$$

Answer 2

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

$$\lim_{a \to \infty} \int_{-a}^a \sin(t)\,dt=0.$$