integral related to arc sine
We want to evaluate the integral
(1) |
Therefore we put an extra variable to the integrand and thus get the function
and in to obtain a simpler integral, we differentiate it under the integral sign (http://planetmath.org/DifferentiationUnderIntegralSign), then integrate:
The gotten expression implies, since , that
and consequently
Here, the substitution (http://planetmath.org/ChangeOfVariableInDefiniteIntegral) helps, yielding
Accordingly, we have the result
For the convergence, see the French version of http://en.wikipedia.org/wiki/Improper_integralthis article.