Trouble with a line in Goldstein

[maverick_starstrider]

Hey, I'm looking through Goldstein's and I'm looking at equation 3.51 where it basically says

$$\int \frac{dx}{\sqrt{\gamma x^2 + \beta x + \alpha}} = \frac{1}{\sqrt{-\gamma} } arccos \left( - \frac{\beta + 2 \gamma x}{\sqrt{\beta^2 - 4 \gamma \alpha} }\right)$$

Every integral book I look at says it should be what he gets except an arcsin function and the argument is positive not negative. but

$$arcsin(-x) \neq arccos(x)$$

What am I missing?

 

[D H]

What am I missing?

You are missing $\pi/2$, which can be absorbed into the (unspecified) constant of integration.

Note that
$$\frac{d}{dx}\arccos(-x) = \frac{d}{dx}\arcsin(x) = \frac 1{\sqrt{1-x^2}}$$

and that
$$\arccos(-x) - \arcsin(x) = \frac{\pi}2$$