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1.37 Standard substitutions for integrals involving ±a2±x2\sqrt{\pm a^{2}\pm x^{2}}.

Integrand factor  substitution{\hbox{Integrand factor}}\quad\quad{\hbox{substitution}}
a2-x2  x=asinuorx=asechu\sqrt{a^{2}-x^{2}}\quad\quad x=a\sin u\quad{\hbox{or}}\quad x=a\,{\hbox{sech}}\,u
a2+x2  x=atanuorx=asinhu\sqrt{a^{2}+x^{2}}\quad\quad x=a\tan u\quad{\hbox{or}}\quad x=a\sinh u
x2-a2  x=acoshuorx=asecu\sqrt{x^{2}-a^{2}}\quad\quad x=a\cosh u\quad{\hbox{or}}\quad x=a\sec u