I have the following equation \begin{equation} \int_{a_s(\mu_1)}^{a_s(\mu_2)} \frac{d a_s}{\beta(a_s)} = \ln \left(\frac{\mu_1}{\mu_2}\right), \end{equation} with $\beta$ being expanded with known coefficients $(\beta_1 - \beta_4)$ \begin{equation} \beta(a_s) = \beta_1 a_s^2 + \beta_2 a_s^3 + \beta_3 a_s^4 + \beta_4 a_s^5. \end{equation}
With a given $a_s(\mu_1), \mu_1$ and $\mu_2$ I now want to calculate $a_s(\mu_2)$ numerically. I have to implement this in a C++ code, so I cannot use Mathematica.
What are the names of the numerical methods I have to use?
First re-write the relation as a differential equation by differentiating both sides with respect to $\mu_2$. I found, $$ \mu_2 \frac{\partial a_s(\mu_2)}{\partial \mu_2} = -\beta(a_s(\mu_2)) \Rightarrow \mu \frac{\partial a_s}{\partial \mu} = -\beta(a_s) $$ Solving a differential equation using a large number of small steps $\delta\mu$ is a basic technique in most numerics textbooks. The wikipedia page should help you start but a trip to the library might be best. I like Kreyszig, Advanced Engineering Mathematics. The initial condition is $a_s(\mu_1)$ and you will be able to evaluate the $\beta$-function using its expansion.