In my text book when I was studying stochastic process, It was presented to my an example about stochastic process, a "oscillator sinusoidal with random phase": $$x(t) = A\cdot sin(2\pi f_0 t + \theta) $$
where $\theta$ is a random variable with uniform distribution from $[- \pi,\pi]$ in other words: $$ p_\theta(\Theta) = \frac{1}{2\pi} \space for [-\pi,\pi]$$
the objective here is discover the correspondent density probability of x(t), so second my mind, I need use this property:
$$p_{x_t}(X) =\frac{ p_\theta(\Theta)}{|g{'}(\theta)|}\Big|_{\theta = g^{-1}(x_t)} \ $$
second my reasoning: $$ |g{'}(\theta)|\Big|_{\theta = g^{-1}(x_t)} \ = (sin^{-1}(x_t) - 2\pi f_0t){'}\Big|_{\theta = g^{-1}(x_t)} \ = \frac{1}{\sqrt{A^{2}-x^{2}}} $$
and finally:
$$ p_{x_t}(X) =\begin{cases} \displaystyle \frac{1}{2\pi\sqrt{A^{2}-x^{2}}} \space\space for \space\space|x| \leq A\\ 0 \space\space\space\space for \space\space |x| > A \end{cases} $$
however, in my text, the result is this: $$ p_{x_t}(X) =\begin{cases} \displaystyle \frac{1}{\pi\sqrt{A^{2}-x^{2}}} \space\space for \space\space|x| \leq A\\ 0 \space\space\space\space for \space\space |x| > A \end{cases} $$ What did I do wrong? I review my calculus a lot of times, and it looks right but I know that is something wrong because the integration from $-\infty$ to $\infty$ is not 1 in the case of my answer, in the case of text book's answer this property is satisfied.