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1
1 Further Integration
1.37
Standard substitutions for integrals involving
±
a
2
±
x
2
\sqrt{\pm a^{2}\pm x^{2}}
.
1.39
Useful identities
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1.38
Substitutions
factor
substitution
derivative
simplification
{\hbox{factor}}\quad\quad{\hbox{substitution}}\quad{\hbox{derivative}}\qquad{% \hbox{simplification}}
a
2
-
x
2
x
=
a
sin
u
d
x
d
u
=
a
cos
u
a
2
-
x
2
=
a
2
cos
2
u
\sqrt{a^{2}-x^{2}}\quad\quad x=a\sin u\qquad{{dx}\over{du}}=a\cos u\quad a^{2}% -x^{2}=a^{2}\cos^{2}u
a
2
+
x
2
x
=
a
tan
u
d
x
d
u
=
a
sec
2
u
a
2
+
x
2
=
a
2
sec
2
u
\sqrt{a^{2}+x^{2}}\quad\quad x=a\tan u\qquad{{dx}\over{du}}=a\sec^{2}u\quad a^% {2}+x^{2}=a^{2}\sec^{2}u
x
2
-
a
2
x
=
a
cosh
u
d
x
d
u
=
a
sinh
u
x
2
-
a
2
=
a
2
sinh
2
u
.
\sqrt{x^{2}-a^{2}}\quad\quad x=a\cosh u\quad{{dx}\over{du}}=a\sinh u\quad x^{2% }-a^{2}=a^{2}\sinh^{2}u.