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

Round your answer to 3 decimal places.

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