Question

Integrate (1)/(sqrt(1-x^2)) from -1 to 1?

Answer 1

`int_-1^1 1/sqrt(1-x^2)dx=pi`

Explanation:

`I=int_-1^1 1/sqrt(1-x^2)dx`
Let `x=sinu` so that `dx=cosudu`.
By using this substitution, we need to change the upper and lower bounds of the integral accordingly:
`-1=sinu` so that `u=-pi/2`
`1=sinu` so that `u=pi/2`
`I=int_(-pi/2)^(pi/2) 1/sqrt(1-sin^2u)cosudu`
Since `sin^2x+cos^2x=1`, `1-sin^2x=cos^2x` and we get
`I=int_(-pi/2)^(pi/2) 1/sqrt(cos^2u)cosudu`
`I=int_(-pi/2)^(pi/2)du`
`I=u|_(-pi/2)^(pi/2)`
`I=pi/2-(-pi/2)`
`I=pi`