Arcsin(1) has many solutions. Which one to use when evaluating definite integral of derivative of arcsin, arccos, etc.?

Question

Kind of a simple question but I don't know how to phrase it well enough to find it using google.

So if I have a definite integral like $$\int^{1}_{-1} \frac{1}{\sqrt{1-x^2}}\, dx$$ I know that the solution is $\arcsin(1)-\arcsin(-1)$ but would that be $\frac{\pi}{2}-\frac{3\pi}{2}$? Or $\frac{\pi}{2}-\frac{-\pi}{2}$? Essentially it's $k\pi$ for some $k$ but what $k$?

I always thought it was just the value between $0$ and $2\pi$, but was recently working on a problem, and using this interpretation gave me a negative volume.

Answer 1

The answer is $\arcsin(1)-\arcsin(-1)=\frac\pi2-\left(-\frac\pi2\right)=\pi$. There is no ambiguity here: $\arcsin\colon[-1,1]\longrightarrow\left[-\frac\pi2,\frac\pi2\right]$ is the inverse of the restriction of the sine function to $\left[-\frac\pi2,\frac\pi2\right]$.

Answer 2

No. By definition $\arcsin$ takes its values in the interval $\Bigl[-\frac\pi2,\frac\pi2\Bigr]$, and the value of the integral is $\color{red}{\pi}$.