I recently came across this property.
Let $f(x)$ and $g(x)$ be two continuous functions in the interval $[a,\,b]$. Then:
$$\left(\int_a^bf(x)\cdot g(x)dx\right)^2=\int_a^bf(x)^2dx+\int_a^bg(x)^2dx$$ I am unable to demonstrate it, and I can't find a demonstration on the internet either. My Thomas' calculus book didn't help.
I saw this expression wrote on the footnote of an old calculus book (the note was written by someone else, not by the author) and now I'm a bit intrigued about its validity. Does this always hold?
A hint or two will be greatly appreciated. Thanks!