Integrate sqrt(1-x^2)

**kinhew93** · January 9th 2013, 03:16 PM

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

**Plato** · January 9th 2013, 04:23 PM

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?

**Soroban** · January 9th 2013, 05:20 PM

Hello, kinhew93!

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

It should be obvious that this requires Trig Substitution.

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

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

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

Can you continue?

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