Assume $\displaystyle \int a(x) b(x) dx = 0$ and $\displaystyle a(x) \neq 0$.

Can this be simplified to $\displaystyle \int b(x) dx = 0$?

Or in another way, if

$\displaystyle \int f(x) g(x) dx = \int f(x) h(x) dx $,

then does this hold:

$\displaystyle \int g(x) dx = \int h(x) dx $