The function $F\left(x\right)=\underset{0}{\overset{x}{\int }}\log (t+\sqrt{1+t^2})dt$  is

Solution we know that if  $f\left(t\right)$ is an odd function, then
$F\left(x\right)=\underset{0}{\overset{x}{\int }}f\left(t\right)dt$ is an even function
Since $\log (t+\sqrt{1+t^2})$  is odd,
$\therefore$ then given integral is even