Dictionary of Engineering Episode 3 Part 7 docx

Appendix
General rules of integration* (cont.)
Ύ
sech
uduϭ sin
Ϫ1
(tanh
u)or2tan
Ϫ1
e
u
Ύ
du
u
2
Ϫ a
2
ϭ
1
2a
ln
΂
u Ϫ a
u ϩ a
΃
ϭϪ
1
a
coth
Ϫ1
u

a
u
2
Ͼ a
2
Ύ
csch
uduϭ ln tanh
u
2
or Ϫcoth
Ϫ1
e
u
Ύ
du
a
2
Ϫ u
2
ϭ
1
2a
ln
΂
a ϩ u
a Ϫ u
΃
ϭ
1

a
tanh
Ϫ1
u
a
u
2
Ͻ a
2
Ύ
sech
2
uduϭ
tanh u
Ύ
du
Ί
a
2
Ϫ u
2
ϭ sin
Ϫ1
u
a
Ύ
csch
2
uduϭϪ
coth u

Ύ
du
Ί
u
2
ϩ a
2
ϭ ln(u
ϩ
Ί
u
2
ϩ a
2
) or sinh
Ϫ1
u
a
Ύ
tanh
2
uduϭ
u Ϫ tanh
u
Ύ
du
Ί
u
2
Ϫ a

2
ϭ ln(u
ϩ
Ί
u
2
Ϫ a
2
)
Ύ
coth
2
uduϭ
u Ϫ coth
u
Ύ
du
u
Ί
u
2
Ϫ a
2
ϭ
1
a
sec
Ϫ1
Ϳ
u

a
Ϳ
Ύ
sinh
2
uduϭ
sinh 2u
4
Ϫ
u
2
ϭ
1
2
(sinh u cosh
u Ϫ u)
Ύ
du
u
Ί
u
2
ϩ a
2
ϭϪ
1
a
ln
΂
a ϩ

Ί
u
2
ϩ a
2
u
΃
Ύ
cosh
2
uduϭ
sinh 2u
4
ϩ
u
2
ϭ
1
2
(sinh u cosh
u ϩ u)
Ύ
du
u
Ί
a
2
Ϫ u
2
ϭϪ

1
a
ln
΂
a ϩ
Ί
a
2
Ϫ u
2
u
΃
Ύ
sech
u tanh uduϭϪ
sech
u
Ύ
f
(n)
gdxϭ
f
(nϪ1)
g Ϫ f
(nϪ2)
gЈ ϩ
f
(nϪ3)
gЉ Ϫ иии
(Ϫ1)

n
Ύ
fg
(n)
dx
Ύ
csch
u coth uduϭϪ
csch
u
This is called generalized integration by parts.
Ύ
du
u
2
ϩ a
2
ϭ
1
a
tan
Ϫ1
u
a
*Here,
u, v, w
are functions of
x;
a, b, p,
q, n any constants, restricted if indicated;

e
ϭ 2.71828 . . . is the natural base of logarithms;
ln
u denotes the natural logarithm of
u where it is
assumed that
u Ͼ
0 [in general, to extend formulas to cases
where u
Ͻ 0 as well, replace ln
u by ln
ȊuȊ]; all angles are in radians; all constants
of integration are omitted but implied.
SOURCE
: Murray R. Spiegel and John Liu,
Mathematical Handbook of Formulas and Tables,
2d ed., Schaum’s Outline Series, McGraw-Hill,
1999.
638
Appendix
Schematic electronic symblols*
*From S. Gibilisco, The Illustrated Dictionary of Electronics, 8th ed., McGraw-Hill, 2001.
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Appendix
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Appendix
641
Appendix
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Appendix

643