# Integration of a fraction.

1. Oct 13, 2007

### Sleek

1. The problem statement, all variables and given/known data

$$\displaystyle \int{\frac{dx}{a^2+\left(x-\frac{1}{x} \right)^2}}$$

2. Relevant equations

-

3. The attempt at a solution

This one looks a bit odd. Had the denominator been a^2 + x^2, it is in one of the standard forms, whose integral is $$\frac{1}{a} \atan{\frac{x}{a}}$$. But the denominator is in the form of a^2 + u^2 (where u is a function of x). I did try some manipulations, but to no avail. I tried putting x as sin(theta), but got something like cos(theta)d(theta)/(a^2+cos^4(theta)/sin^2(theta)), which seems even more complex. If someone can just point me into the direction to look, I'll attempt the solution.

Thank you,
Sleek.

2. Oct 13, 2007

### Gib Z

Expand the brackets, simplify, multiply the entire integral by x^2/x^2, factor the denominator and partial fractions.

3. Oct 13, 2007

### Sleek

Thanks for the quick reply, I'm currently here,

$$\displaystyle \int{\frac{x^2dx}{x^2(a^2-2)+x^4+1}$$

I don't see how I can factorize/simplify the denominator or the expression...?

Regards,
Sleek.

Last edited: Oct 13, 2007
4. Oct 13, 2007

### Gib Z

Well let $a^2-2 =b$ and $u=x^2$. Now it resembles a nice quadratic equation =]