What happens if we try the substitution
y = x2 − 4?
The transformation formula for differentials is
dy = 2x⋅dx and the
expression on the right is not in the integral, which does not sound too
optimistic. When we try to rearrange the differential transformation
formula, we get this.
Our suspision is vindicated, the new integral is not any better than the
given one.
Now we will try out our favorite mixed substitution for removing roots, that
is,
y2 = x2 − 4.
Differentials get transformed using the formula
2y⋅dy = 2x⋅dx
and again we have to tweak it a bit.
Again, this was not any help.