# My Fifteenth Problem

Calculus Level 5

Let $$f: \left [ \dfrac \pi2 , \pi \right ] \mapsto \left [ \dfrac12 , 2\right ]$$ be the bijection $$f(\theta) = \dfrac{1-\cos\theta}{1+\sin\theta}$$.

And let $$g : \left [ -\dfrac12 , \dfrac12 \right ]\mapsto \left [ \dfrac12 , 2\right ]$$ be the injection $$g(x) = e^{\arcsin x}$$.

Evaluate $$\displaystyle \large \int_{-1/2}^{1/2} f^{-1} \circ g(t) \, dt$$.