TITLE: Upper bound of $\int_{-\infty}^{\infty}\sin(x)dx$.
QUESTION [1 upvotes]: From another question, improprer integral
$$\int_{-\infty}^{\infty}\sin(x)dx$$
is not
$$\lim_{a \to \infty} \int_{-a}^a \sin x \, d x$$
but
$$\lim_{a \to \infty}\lim_{b \to \infty}\int_{-a}^b \sin x \, d x.$$
This is also valid for
$$\lim_{a \to \infty}\lim_{b \to \infty}\int_{-a}^b \cos x \, d x.$$
For this reason there is no limit. However, is it possible to obtain some estimations (from above) of the following?
$$\int_{-a}^b \cos x \, d x, \ \ \int_{-a}^b \sin x \, d x$$
$$\left|\int_{-a}^b \cos x \, d x\right|, \ \ \left|\int_{-a}^b \sin x \, d x\right|$$
I think that the upper bound of these integrals is $4$.

REPLY [3 votes]: Simply use 
$$
\bigg|\int_a^b \cos (x)dx \bigg|=|\sin (b)-\sin (a)|\leq 2.
$$
By choosing $b = \pi/2$ and $a=-\pi/2$, one can see that this bound cannot be improved. The calculation for $\sin $ is analogous.
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