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Sometimes you are looking for deeper insights in some equations that can only be achieved by finding symbolic solutions.

You would never figure out that black holes are a solution to the Einstein field equations of you just threw a numerical solver at it, for example. (Bad example because that's arguably the easiest solution to them but I hope you get my point.)

0.01% would whip out a numerical solver; 99.98% would go "huh?"

It's going to take you a while to numerically solve that integral for the uncountably many values of \alpha that you are being asked to...

It's probably best to return a function that takes alpha as a parameter and numerically integrates for whichever finite set of alphas are required by the caller.

The usual method is to computer a reasonable number of solutions and do curve fitting.

It's the algorithm in and of itself that serves as the solution to the integral.