Integrating Inverse Algebraic Equations

Is there a more intuitive and logical way to integrate an inverse algebraic equation (i.e. 1/(x^2 + 4x + 29)) than factorising it and using an identity? The way it's done in school is to factorise the equation, and then use a "shortcut" from the tables (i.e. it becomes 1/a tan^-1 x/a). That way isn't hard to do at all, but, I'm wondering is there a more logical and intuitive way to do it, a way that you know what you're actually doing?

I've been trying to do it using a substitution, differentiating the substitution twice (that way I get 1/2 du^2 = dx^2), then messing around a few ways using double integrals. But, I can't seem to get it.

So is there a better way?

Thanks.

Cokehead Mother

Okay I'm not really sure if this is the sort of thing you're looking for, but I'd use the same technique you use in LC to integrate root(a^2 - x^2).

Your integral is int dx/[(x + 2)^2 + 5^2]

5^2tan^2(u) + 5^2 = 5^2sec^2(u)

and 5tan(u) differentiates to 5sec^2(u)

So if you make the substitution x + 2 = 5tanu, it comes out nicely.