General question about solving equations involving a definite integral

Question

Are there any well known techniques to solve a problem of the following form: $$\int_a^b f(x,\alpha) dx = g(\alpha),$$ where $a,b\in\mathbb{R}$ are fixed, $f$ and $g$ are known functions, $\alpha\in\mathbb{C}$ is the unknown variable, and the expression is not an identity. Put another way, given the above expression are there techniques available to find the values of $\alpha$ for which the expression holds true, assuming we know from empirical study that there do exist such $\alpha$ ?

Answer 1

It's a question slightly strange, but under certain not-too-tight conditions, we have $$\frac{d}{d\alpha}\left(\int_a^bf(x,\alpha)dx\right)=\int_a^b\frac{d}{d\alpha}\left(f(x,\alpha)\right)dx$$ So if you know both functions you could check whether $\,g'(\alpha)\,$ equals the above...