If

[tex]\int_0^\infty f(s)ds=\int_0^\infty g(s)ds[/tex]

What are sufficient conditions such that [tex]f(s)=g(s)[/tex]?

I know that two functions [tex]f(s),g(s)[/tex] are equal if their domain, call it [tex]S[/tex], is equal and if [tex]f(s)=g(s)[/tex] for all [tex]s\in S[/tex] but I can't figure this one out.

The full problem is actually

[tex]-\int_0^\infty\frac{d}{ds}\left(f(\vec{r}+s\hat{\Omega},E,\hat{\Omega})e^{-\Sigma_ts}\right)ds[/tex]

[tex]=\int_0^\infty\Sigma_te^{-\Sigma_ts}g(\vec{r}+s\hat{\Omega},E,\hat{\Omega})ds[/tex]

and therefore hopefully

[tex]-\frac{d}{ds}\left(f(\vec{r}+s\hat{\Omega},E,\hat{\Omega})e^{-\Sigma_ts}\right)=\Sigma_te^{-\Sigma_ts}g(\vec{r}+s\hat{\Omega},E,\hat{\Omega})[/tex]

Please help me!!! I'm finishing soon with my work and proving this (or at least having sufficient conditions) would be very welcome!