Converting a singular integral into a non-singular integral

Question

The integral I am working with is $$\int_0^{50 \pi/180} (\cos(x)-\cos(50\pi/180))^{-1/2} dx$$ I am trying to convert this singular integral to a non-singular integral. Is there a way to change the variable to make this non- singular so I can solve this?

Answer 1

If $a = 50 \pi/180$ is the upper endpoint of the integral,

$$ \lim_{x \to a} \dfrac{\cos(x) - \cos(a)}{x - a} = \cos'(a) = -\sin(a) \ne 0$$ so the integrand goes to infinity as $x \to a-$, but only like $(x-a)^{-1/2}$ which is integrable. This suggests the change of variables $a - x = t^2$ which makes the integral into $$ \int_0^{\sqrt{a}} \dfrac{t\; dt}{\sqrt{\cos(a-t^2) - \cos(a)}}$$

BTW, your integral has a "closed form" expression in terms of elliptic integrals. In Maple's notation (which follows Gradshteyn and Ryzhik), it is

$$ \sqrt {2}{\it EllipticF} \left( 2\,{\frac {\cos \left( {\frac {13\,\pi }{36}} \right) }{\sqrt {-2\,\cos \left( {\frac {5\,\pi}{18}} \right) + 2}}},1/2\,\sqrt {-2\,\cos \left( {\frac {5\,\pi}{18}} \right) +2} \right) $$ where $EllipticF$ is the incomplete elliptic integral of the first kind.