Absolute value in indefinite integral

Question

I have to show that $$ \int \left(\frac{dx}{x^2 \sqrt{x^2+4}}\right) = \left(\frac{-\sqrt{x^2+4}}{4x}\right) + c$$

I used the substitution $ \frac{x}{2} = \tan u$, and I got: $$\frac{1}{4}\int \left(\frac{ |\cos u|\; du}{ (\sin u)^2 }\right)$$

I saw in the solution of this task that $|\cos u| = \cos u$. Why do we ignore the absolute value?

Thanks in advance.

Answer 1

Note that the domain of $\tan (u)$ is $u \in \big(-\frac{\pi}{2}, \frac{\pi}{2}\big)$.

Where is $\cos(u)$ positive? What can you conclude from that regarding $\vert \cos(u)\vert$ and $\cos(u)$?

Answer 2

When defining $x=\tan u$ and from $$-\infty<x=\tan u<\infty$$we conclude that $$-{\pi \over 2}<u<{\pi \over 2}$$is sufficient. In this interval $0<\cos u\le1$ and $$|\cos u|=\cos u$$