Integral of sqrt(1-x^2)/x^2

[pamw]

Integral of sqrt(1-x^2)/x^2

I see the derivative kinda of arcsin(x) in there and I thought about multiplying by sqrt(1-x^2)/sqrt(1-x^2), but I don't know where to go from there.

Any suggestions? Thank you.

 

[Count Iblis]

Try partial integration. The integral of the factor 1/x^2 is -1/x. The derivative of sqrt[1-x^2] is -x/sqrt[1-x^2] The integral of the product of the two is just an arcsin...

 

[galactus]

This one can be done nicely by trig sub.

Let \(\displaystyle x=sin({\theta}), \;\ dx=cos({\theta})d{\theta}\)

Upon subbing, you have:

\(\displaystyle \L\\\int\frac{\sqrt{1-sin^{2}({\theta})}}{sin^{2}({\theta})}\cdot{cos({\theta})d{\theta}\)

This reduces to:

\(\displaystyle \L\\\int\frac{cos^{2}({\theta})}{sin^{2}({\theta})}d{\theta}\)

\(\displaystyle \L\\\int{cot^{2}({\theta})}d{\theta}\)

After integrating, you can express it in terms of x by using:

\(\displaystyle x=sin({\theta}), \;\ cos({\theta})=\sqrt{1-x^{2}}, \;\ {\theta}=sin^{-1}(x)\)