We know for $ u > 1 $ $$ \int_{-\pi/2}^{+\pi/2} \ln(\sin(x) + u) dx = \pi \left(\ln\left(u + \sqrt{u^2 -1}\right) - \ln(2)\right) $$
Usually this is shown by using differentiation under the Integral sign or contour integration.
This made me wonder :
Consider the log-transform
For a given real continuous function $F(u)$ find a real continuous function $g(x) $ that only depends on $x$ ( not on $u$ ) such that
$$ F(u) = \int_{-\pi/2}^{+\pi/2} \ln(g(x) + u) dx $$
So
$$ \mathcal{L}(F(u)) = g(u) $$
Where $\mathcal{L}$ stands for “ log-transform “.
For instance $$ \mathcal{L}\left[ \pi \left(\ln\left(u + \sqrt{u^2 -1}\right) - \ln(2)\right) \right] = \sin(u). $$
Notice that in this case $\sin(2u),\sin(3u),\sin(4u),... $ and $\sin(-u),\sin(-2u),\sin(-3u),\sin(-4u),... $ are also solutions ! Perhaps uniqueness comes from functions that are strictly increasing ? ( $\mathcal{L} ... = exp(u) $? )
What is known about these (inverse) integral transforms ?
Is there an Integral representation for them ?