Question

I need help with integrating these two problems. Im stuck.

1. integrate (sin^-1)dx/((1-x^2)^3/2)
sin^-1 aka arcsin


2. integrate dx/((1-x^2)^3/2) by using 1/z



Any and all help will be appreciated!

Answer 1

#1 makes no sense

#2 dx/(1-x^2)^(3/2)
If you let
x = 1/z
dx = -1/z^2 dz
1-x^2 = 1 - 1/z^2 = (z^2-1)/z^2

Now you have

-dz/z^2 z^3/(z^2-1)^(3/2)
= -z dz/(z^2-1)^(3/2)

and the integral is simply
1/√(z^2-1) = x/√(1-x^2)