Improper integral $\int_a^bf(x)dx$ is convergent and $g(x)\ge0$. If $\int_a^bg(x)dx=0$, show that $\int_a^bf(x)g(x)dx=0.$ Assume $f(x)\in\mathrm{C}[a,b),\lim_{x\to b}f(x)=+\infty$, $\int_a^bf(x)dx$ is convergent. And the non-negative function $g(x)$ on the interval $[a,b]$ is Riemann-integrable. If $\int_a^bg(x)dx=0$, show that
$$\int_a^bf(x)g(x)dx=0.$$
Since  $\int_a^bf(x)dx$ is an improper integral, the normal Integral mean value theorem can't work. How can I deal with it? 

Hint: $g(x)$ is non-negative on $[a,b]$ and it's integral is zero.
