I am considering the function \(\displaystyle u(t,x)=\sqrt{1+x^{2}+t^{2}}\sin(x)\).

This function is continuous and therefore measurable. Furthermore:

\(\displaystyle \int_{[0,2\pi]}\vert \sqrt{1+x^{2}+t^{2}}\sin(x) \vert \leq \int_{[0,2\pi]} \sqrt{1+x^{2}+t^{2}} \)

How do I continue from here to sho that the above is \(\displaystyle < \infty\)?

Thanks.