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

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  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

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