How do you complete the proof for the integration of dx/sqrt(a^2 - x^2) = arcsin (x/a) + C.

Question

a is a constant.

Answer 1

Substitute x = a sint(t) in

dx/sqrt(a^2 - x^2)

dx = a cos(t) dt

sqrt(a^2 - x^2) = a |cos(t)|

Because x is between -a and a, t can be chose to be between -pi/2 and pi/2, which means that cos(t) is postive, so we can omit the absolute value signs. This means that:

dx/sqrt(a^2 - x^2) = dt

The integral is thus t + C=
arcsin (x/a) + C