Integrate sqrt(1-x^2)

• Jan 9th 2013, 03:16 PM
kinhew93
Integrate sqrt(1-x^2)
I need to integrate sqrt(1-(x^2))

I know that the reciprical of this integrates to arcsin(x) + k but I don't know how I can relate this.

Any help would be much appreciated :)
• Jan 9th 2013, 04:23 PM
Plato
Re: Integrate sqrt(1-x^2)
Quote:

Originally Posted by kinhew93
I need to integrate sqrt(1-(x^2))
I know that the reciprical of this integrates to arcsin(x) + k but I don't know how

Why do you not use web resources like this?
• Jan 9th 2013, 05:20 PM
Soroban
Re: Integrate sqrt(1-x^2)
Hello, kinhew93!

Quote:

$\displaystyle I \;=\; \int \sqrt{1-x^2}\,dx$

It should be obvious that this requires Trig Substitution.

Let $\displaystyle x \,=\,\sin\theta \quad\Rightarrow\quad dx \,=\,\cos\theta\,d\theta$

. . And: .$\displaystyle \sqrt{1-x^2} \:=\:\sqrt{1-\sin^2\!x} \:=\:\sqrt{\cos^2\!x} \:=\:\cos x$

Substitute: .$\displaystyle I \;=\;\int\cos\theta(\cos\theta\,d\theta) \;=\;\int\cos^2\!\theta\,d\theta \;=\;\tfrac{1}{2}\int (\cos2\theta - 1)\,d\theta$

Can you continue?