Show integrability

Sep 2016
15
0
Denmark
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
Nov 2013
6,647
2,994
California
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

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