How do I demonstrate the truth of this integral equality?

Question

How do I show that the integral of du / sqrt (a^2 - u^2) equals inverse sin of u/a + C?

Answer 1

I = ∫du / sqrt (a^2 - u^2)

let us substitute u = a sin (p) or p = sin^-1 (u/a) ----(1)
so du = a cos (p) dp

I = ∫a cos (p) dp / sqrt {a^2 cos^2 (p)}

I = ∫a cos (p) dp / {a cos (p)} = ∫dp integrating

I = p +C putting back value of p from (1)

I = sin^-1 (u/a) +C answer

equals inverse sin of u/a + C?

Answer 2

Start by saying let u = a*sinx then find du/dx and substitute both into the integral.

Answer 3

I am not tying to be flip, but buy Schaum's Outline Series
Calculus
Integral and Differential Calculus
therein, you will find the answer with lots of examples.