How Do You Integrate \( \frac{x}{(1-x^2)} \) dx?

[tomwilliam2]

Homework Statement

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

Homework Equations

Integration by parts, by substitution, etc.

The Attempt at a Solution

I just can't remember how to begin this integration. I tried doing integration by parts, where
$$a(x) = x$$
$$a'(x) = 1$$
$$b(x) = $$
$$b'(x) = \frac{1}{(1-x^2)}$$
$$\int \frac{x}{(1-x^2)}dx = xb(x) - \int \frac{1}{(1-x^2)} dx$$

But couldn't work out what form b(x) should take.
Thanks in advance for any help

 

[Fightfish]

If you have seen enough of these integrals, they can essentially done "by observation"

If not, substitution is a good method to use. Substitute $y = x^{2}$. Then $dy = 2x dx$
Then the integral becomes:
$$\int \frac{1}{2}\frac{1}{1-y}dy$$

Does that now look more doable?

 

[HallsofIvy]

Why not let u= 1- x^2 itself? Then du= -2xdx so that xdx= -(1/2)du do that
$-\frac{1}{2}\int \frac{du}{u}$

 

[tomwilliam2]

Ah yes, thanks...that makes sense. I wasn't getting where the sqrt cam from in the first response.
Thanks to both of you

 

[Fightfish]

I wasn't getting where the sqrt cam from in the first response

Oops, typo, my bad.