I tried the substitution but got for the second bracket one plus k squared times x squared, and why did the G term disappear. Thank you.
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First, the reason is that it's an integral of the form $\sqrt{G.....}$, so the G is just a coefficient and nonessential, so they just omitted it in the next line, because they were focusing on the substitution.
Also, there is a negative sign because they are substituting $\displaystyle k^2 = -\frac{n}{m}$. Perhaps you may have accidentally forgotten to include the negative.
Saketh Malyala
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The minus n/m becomes positive k^2 – mathExplorer Jun 07 '19 at 03:01
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But there's also a minus sign coming in from the $\sqrt{-m}$. I think the author meant to write $k^2=n/m$. – Gerry Myerson Jun 07 '19 at 03:01
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2I suspected that as well thank you confirming :) – mathExplorer Jun 07 '19 at 03:02
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both of those are possibilities, just check that it doesn’t cause a dramatic difference in the final result – Saketh Malyala Jun 07 '19 at 03:12