Integrating 1/(x - sqrt(1-x^2))

[haleycomet2]

Homework Statement

$\int$$\frac{dx}{x-\sqrt{1-x^2}}$

Homework Equations

The Attempt at a Solution

i try to simplify it by multiplying $\frac{x+\sqrt{1-x^2}}{x+\sqrt{1-x^2}}$,becoming =$\frac{x+\sqrt{1-x^2}}{2x^2-1}$
=$\int${$\frac{x}{2x^2-1}$+$\frac{\sqrt{1-x^2}}{2x^2-1}$}dx

ps:
a.Can i partial fraction the last term to $\frac{A}{x-\frac{1}{\sqrt{2}}}$+$\frac{B}{x+\frac{1}{\sqrt{2}}}$??
b.i try to integrate by using wolfram alpha online ,but the steps is incredibly long..
does it exist any other simpler way??

 

[Char. Limit]

I checked W-A, and they did it the same way I would have: Begin with a trig substitution like x=sin(u), then use the Weierstrass substitution: v=tan(u/2). It will be long and difficult, but it's possible.

 

[eumyang]

a.Can i partial fraction the last term to $\frac{A}{x-\frac{1}{\sqrt{2}}}$+$\frac{B}{x+\frac{1}{\sqrt{2}}}$??

No, because both the numerator and denominator of the original fraction must be polynomials, and in
$$\frac{\sqrt{1-x^2}}{2x^2-1}$$
the numerator is not a polynomial.

 

[haleycomet2]

o,i see.Thanks a lot