Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.
Sign up to join this community
Given this integral
$$\int_{-\infty}^{+\infty}\sin(\cosh x)\cos(\sinh x)\mathrm dx={\pi\over 2}\tag1$$
My try:
Recall $$2\sin(A)\cos(B)=\sin(A-B)+\sin(A+B)$$
$(1)$ becomes
$$\frac12\int_{-\infty}^{+\infty}\cos(\cosh x-\sinh x)+\sin(e^x)\mathrm dx\tag2$$
Recall $$\cosh^2 x-\sinh^2 x=1$$
$(2)$ becomes
$$\frac12\int_{-\infty}^{+\infty}\sin(e^{-x})+\sin(e^x)\mathrm dx\tag3$$
How may be prove $(1)?$
The correct prosthaphaeresis formula is $$ 2\sin{A}\cos{B} = \sin{(A+B)}+\sin{(A-B)}. $$ This gives $$ \frac{1}{2} \int_{-\infty}^{\infty} (\sin{(e^{-x})} + \sin{(e^{x})}) \, dx, $$ from the definitions of the hyperbolic functions, $$ \cosh{x}+\sinh{x} = e^x, \qquad \cosh{x}-\sinh{x} = e^{-x}. $$ The integrand is an even function, so this is the same as $$ \int_{-\infty}^{\infty} \sin{(e^{x})} \, dx $$ by putting $y=-x$ in one of the terms. Now put $y=e^{x}$. The limits change to $0$ and $\infty$, and $dx=dy/y$, so the integral becomes $$ \int_0^{\infty} \frac{\sin{y}}{y} \, dy, $$ which can be done in numerous ways.
