Consider the integral $$ \int \limits_{-1}^{1}d(\cos(\theta)) = 2 $$ I want to change the variable by the rule $$ \tag 1 \cos(\theta') = \frac{E'_{N}(\theta)p_{B} - m_{B}E_{N}}{p_{B}\sqrt{(E'_{N}(\theta))^{2}-m_{N}^{2}}}, \quad E_{N}'(\theta) =\frac{\sqrt{p_{B}^{2}+m_{B}^{2}}}{m_{B}}\left(E_{N} + \frac{p_{B}}{\sqrt{p_{B}^{2}+m_{B}^{2}}}\sqrt{E_{N}^{2} - m_{N}^{2}}\cos(\theta)\right) $$ Here the parameters $m_{B}, m_{N}, p_{B},E_{N}$ are real positive numbers with $E_{N}\gtrsim m_{N}$.
It can be shown using $(1)$ that if $1-\frac{m_{B}^{2}(E_{N}^{2}-m_{N}^{2})}{m_{N}^{2}p_{B}^{2}}>0$, then the domain of the definition of $\cos(\theta')$ is $$ \tag 2 \cos(\theta')\in \left(\sqrt{1-\frac{m_{B}^{2}(E_{N}^{2}-m_{N}^{2})}{m_{N}^{2}p_{B}^{2}}};1 \right), $$ while otherwise it reads $$ \tag 3 \cos(\theta') \in (-1,1) $$
My question is the following. The inverse function to $(1)$, which is $\cos(\theta) = f(\cos(\theta'))$, is two-valued, and the whole domain $\cos(\theta) \in (-1,1)$ is covered only by these two "branches" $f_{1,2}(\cos(\theta'))$. Namely, introducing "jacobians" $|df_{i}(\cos(\theta'))/d(\cos(\theta'))|$ I have $$ \tag 4 \int \limits_{\cos(\theta'_{\text{min}})}^{1}\left|\frac{df_{1}}{d\cos(\theta')}\right|d(\cos(\theta'))+\int \limits_{\cos(\theta'_{\text{min}})}^{1}\left|\frac{df_{2}}{d\cos(\theta')}\right|d(\cos(\theta')) = 2 $$ for the domains $(2)$, and $$ \tag 5 \int \limits_{-1}^{1}\left|\frac{df_{1}}{d\cos(\theta')}\right|d(\cos(\theta'))+\int \limits_{-1}^{1}\left|\frac{df_{2}}{d\cos(\theta')}\right|d(\cos(\theta')) = 4 $$ for the domain $(3)$ (each particular term isn't equal to 2)!
I don't understand how to calculate jacobian for the given substitution correctly and how to use two branches for the integration of arbitrary functions $g(\theta(\theta'))$, say, simple $\cos(\theta)$. The sums $(4),(5)$ look strange as for me, I can't give an explanation why they give correct result. Could you please help me?
Edit. Using (4), (5) I can guess that
$$ \int \limits_{-1}^{1}g(\cos(\theta))d\cos(\theta) = \int \limits_{\cos(\theta'_{\text{min}})}^{1}g(f_{1}(\cos(\theta')))\left|\frac{df_{1}}{d\cos(\theta')}\right|d\cos(\theta')+\int \limits_{\cos(\theta'_{\text{min}})}^{1}g(f_{2}(\cos(\theta')))\left|\frac{df_{2}}{d\cos(\theta')}\right|d\cos(\theta'), $$ but I don't understand the math...
See also the picture.