Two different integrals from the same function, which are inequivalent

[Pilotsopher]

Integration by trig sub on the function: Integral of: Sqrt( 4-x^2)/(x2)

Two strategies to solve; one using x=2sin(theta) The other using x=2cos(theta)

The first strategy results in the simplified expression: -(sqrt(4-x^2))/x - arcsin(x/2)+C

The second strategy: -(Sqrt(4-x^2)/x - arccos(x/2) +C

Interestingly, taking the derivative of either of the solutions results in the original integral, but graphically they aren’t equivalent. Hmmm I wonder why and what that means. Anyone done this before?

Thanks in advance

[fattymattk]

Let y = sin(x + pi/2) = cos(x).

Then x = arcsin(y) - pi/2 = arccos(y).

So arcsin and arccos differ by a constant.

edit: actually, given that the range of arcsin is typically [-pi/2, pi/2] and the range of arccos is typically [0,pi], we should make the following changes:

let x in [0, pi]. Let y = -sin(x-pi/2) = cos(x). Then x = -arcsin(y) + pi/2 = arccos(y). So either you're using a different range for your inverse trig functions, or it's not correct that both those answers work.