Calculus – How to Calculate ?(cos(x)sin(?(1+x²))/?(1+x²)) dx

Question

$$
\int_{0}^{+\infty} \frac{\cos x \sin \sqrt{1+x^{2}}}{\sqrt{1+x^{2}}} \mathrm{~d} x
$$
My idea: let $x = \sinh u$ and $\sqrt{1+x^2} = \cosh u$ then the formula simplified as
$$
\int_{0}^{+\infty} \cos({\sinh u)\sin(\cosh u)} \mathrm{~d} u
$$
I use wolframalpha to get
$$
\begin{align}
&\int\cos({\sinh u)\sin(\cosh u)} \mathrm{~d} u \\
&= 1/2 (-Si(\cosh(u) – \sinh(u)) + Si(\cosh(u) + \sinh(u)))
\end{align}
$$
where $Si(z) = \int_{0}^{z} \frac{\sin x}{x}\mathrm{d}x$
Is there any other answer ? Thank you.

Answer 1

Continue with

\begin{align} &\int_{0}^{+\infty} \cos({\sinh u)\sin(\cosh u)} {~d} u\\ = &\frac12 \int_{0}^{+\infty} (\sin e^u + \sin e^{-u})du = \frac12 \int_{-\infty}^{+\infty} \sin e^u du\overset{t=e^u}=\frac12\int_{0}^{+\infty}\frac{\sin t}t dt = \frac\pi4 \end{align}