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1 Further Integration
1.36
Hyperbolic triangles
1.38
Substitutions
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1.37
Standard substitutions for integrals involving
±
a
2
±
x
2
\sqrt{\pm a^{2}\pm x^{2}}
.
Integrand factor
substitution
{\hbox{Integrand factor}}\quad\quad{\hbox{substitution}}
a
2
-
x
2
x
=
a
sin
u
or
x
=
a
sech
u
\sqrt{a^{2}-x^{2}}\quad\quad x=a\sin u\quad{\hbox{or}}\quad x=a\,{\hbox{sech}}\,u
a
2
+
x
2
x
=
a
tan
u
or
x
=
a
sinh
u
\sqrt{a^{2}+x^{2}}\quad\quad x=a\tan u\quad{\hbox{or}}\quad x=a\sinh u
x
2
-
a
2
x
=
a
cosh
u
or
x
=
a
sec
u
\sqrt{x^{2}-a^{2}}\quad\quad x=a\cosh u\quad{\hbox{or}}\quad x=a\sec u