I am stuck on part of my aerodynamics textbook which involves an integral equation. I would like some help understanding how to solve the equation.
Problem Statement
Find $\gamma(\theta)$ such that $\gamma(\pi)=0$ and
$$\int_0^\pi\frac{\gamma(\theta)\sin\theta}{\cos\theta-\cos\theta_0}d\theta=A$$
where $A$ is some constant real number and $\theta_0 \in(0,\pi)$.
Extra Background
I am currently taking a course in Aerodynamics with the textbook "Fundamentals of Aerodynamics" by John D. Anderson.
In Chapter 4, the author presents classical thin airfoil theory, where the goal is to find the proper vortex strength distribution $\gamma(x)$ along the camber line of the airfoil. Through some mathematical manipulation, this is expressed as $$\frac{1}{2\pi}\int_0^c\frac{\gamma(\xi)d\xi}{x-\xi}=V_{\infty}\alpha$$
where $x$ is some number in $(0,c)$, and $V_\infty\alpha$ is a constant, and the function $\gamma(x)$ satisfies $\gamma(c)=0$. Making the substitution $\xi=\frac{c}{2}\left(1-cos\theta\right)$, the integral can be rewritten $$\frac{1}{2\pi}\int_0^\pi\frac{\gamma(\theta)\sin\theta d\theta}{\cos\theta-\cos\theta_0}=V_{\infty}\alpha$$ The author then claims that this equation has solution $$\gamma(\theta)=2\alpha V_\infty\frac{1+\cos\theta}{\sin\theta}$$ but does not describe why this is the case.