Solving a double integral numerically

I am working on a research paper and need to solve the below eqation numerically. As there is no analytical solution for the question, I need to solve it numerically. Can anyone help me please?

[David Hill]

You want to solve the ODE for T=what? What are the constants (sigma, k0,beta,R,E0)? What is your boundary condition, alpha(0)=?

[Ayush Kumar A]

The value of T lies from 300K to 1200K. So, I would be getting an array containing all the alpha values for this range. Sigma lies from 5 to 70, beta = 10,the values of k0 are between 104 and 1030,for E0 are between 50 and 350. R is the universal gas constant.

The objective is to find out the best value of E by minimizing the below equation.