# Integral of 1/{x-(1-x^2)0.5}

1. Jun 26, 2011

### haleycomet2

1. The problem statement, all variables and given/known data
$\int$$\frac{dx}{x-\sqrt{1-x^2}}$

2. Relevant equations

3. 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..:yuck:
does it exist any other simpler way??

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2. Jun 26, 2011

### 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.

3. Jun 26, 2011

### eumyang

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.

4. Jun 26, 2011

### haleycomet2

o,i see.Thanks a lot