# Show integrability

#### Meelas

Hey,

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.

#### romsek

MHF Helper
You're only integrating over one of the variables so the other, say $t$ is constant for the purposes of the integration.

$\sqrt{1+t^2+x^2} \leq \sqrt{1+t^2+4\pi^2},~x \in [0,2\pi]$
so

$\displaystyle \int_0^{2\pi}~\sqrt{1+t^2+x^2}~dx \leq 2\pi \sqrt{1+t^2 + 4\pi^2} < \infty$

Meelas