integration - An almost Fresnel integral

$\def\d{\mathrm{d}}$So I tried doing the following integral:

$$I=\int_0^{+\infty}\sin(2^x)\,\d x,$$

which is quite similar to the famous Fresnel integral. First, I rewrote $\sin$ using its complex exponential definition, then I let $u=2^x$:

$$I=\int_0^{+\infty}\frac{e^{i2^x}-e^{-i2^x}}{2i}\,\d x = \frac1{2i\ln(2)} \int_1^{+\infty} \frac{e^{iu}-e^{-iu}}u \,\d u.$$

(so close to letting me use Frullani's integral $\ddot\frown$)

But where do I go from here? It looks very close to a place where I could use the exponential integral or something like that, but not quite...

Answer

Hint. One may perform the change of variable

$$
u=2^x, \quad \ln x= \frac1{\ln 2}\cdot \ln u, \quad dx=\frac1{\ln 2}\cdot \frac{du}u,
$$ giving
$$
I=\int_0^{+\infty}\sin(2^x)\ dx=\frac1{\ln 2}\cdot\int_1^{+\infty}\frac{\sin(u)}{u}\ du=\frac1{\ln 2}\cdot\left(\frac{\pi }{2}-\text{Si}(1)\right)
$$ where we have made use of the sine integral function $\text{Si}(\cdot)$.