Why do we ignore the absolute value in hyperbolic substitutions?

Solution 1:

It is common practice to try substitutions without caring for "mundanities" such as sign discussions, and postpone them, the main challenge being to find an antiderivative.

In some cases, the domain of integration is such that no sign error is possible. In other cases, reverting to original variables may annihilate the sign problem. Fixing signs can also be done in the end, when checking the derivative. So there is no hurry.

In the present case, the integrand is an even function so that the antiderivative will be odd, and solving in the positives is enough (complete solution by mirroring).