How to integrate int 1/(4+x^2)^2 dx

[sandbanana]

Homework Statement

$$\frac{dx}{(4+x^2)^2}$$

Homework Equations

I understand that $$\frac{dx}{a^2+u^2}$$ = $$\frac{1}{a}$$ tan ^-1 $$\frac{u}{a}$$ + c

The Attempt at a Solution

The extra square is throwing me off for some reason. If I let a=2 and u=x it doesn't seem to help because of the whole term being squared. I end up with $$\frac{1}{2}$$tan^-1$$\frac{x}{2}$$ + c. In my book the first fraction is $$\frac{1}{16}$$I'm pretty sure I can't complete the square because it isn't ax^2+bx+c. Basically I am stumped.

 

[Mark44]

I would go with a trig substitution, tan u = x/2.

 

[sandbanana]

Yeah I can't believe I missed that substitution. I was going through my notes and realized I let that square block my train of thought for some reason. Its amazing what a small break and clearing your head can do.
I just let a=2 and x=2 tan theta. What I forgot to do was take the derivative of x. Plugging it all back together and then using some right angle trigonometry I found out the problem was much easier than I though.
Thanks for the help.

 

[n!kofeyn]

This is a good substitution technique to internalize. Often when you have a²+x² in the denominator, you will make the substitution x=a*tan(u). If you have a²-x², then you make either x=a*cos(u) or x=a*sin(u).