indefinite trig substitution integral

Question

I am not able to understand how to get the solution for an integral. Substituting something like $\frac{1}{12}tan(\theta)$ seems to be the right thing to do, but I can't figure it out from there.

$$\int\frac{24dx}{(144x^2+1)^2}$$

Answer 1

Make your substitution $x=\frac{1}{12}\tan(u)$. Then, $dx=\frac{1}{12}sec^2(u)du$, while the denominator becomes $(\tan^2(u)+1)^2=\sec^4(u)$. Thus, you have to integrate $2\cos^2(u)=(1+\cos(2u))$.