# Double Substitution Correct or Faulty?

1. Mar 7, 2014

### Nano-Passion

Hey. I was reviewing some trig substitution and I know how to solve the problem but I was curious if you can also solve it by using two different substitutions/change of varaibles? If so, how would you relate the functions? Consider for example:

∫ (1-x^2)^1/2 dx, -1<= x <= 1

x = sin Θ
dx = cos Θ dΘ

-pi/2 <= Θ <= pi/2
∫ (1-x^2)^1/2 dx = ∫ 1 - sin^2Θ dΘ

u = 2Θ
du = 2 dΘ

-pi <= u <= pi
Then the integral is equal to:

Θ - 1/2Θ - 1/2 sin u(Θ(x)) + C

where Θ(x) = sin^-1x.

Would u(Θ(x)) = 2arcsinx be true?

Which would give us:

arcsin x - 1/2arcsin x - 1/2 sin (2arcsin x) + C

However ugly it is, would this be correct with the adjusted interval? Does it make sense?

2. Mar 7, 2014

### tiny-tim

Hi Nano-Passion!
sorry, not following this

you've left out the √, and the cos Θ dΘ

(and i think later on you're confusing 2Θ with 2Θ)