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Reference Guide

# Calculus Integrals I - Reference Guides

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INDEFINITE INTEGRALS INTEGRALS INVOLVING u2± a 2
INTEGRALS INVOLVING a2– u 2
BASIC TAB L E O F I N T E G R A L S
Calculus Integrals I
Calculus Integrals I
• If F(u) + Chas the derivative f(u), then F(u) + C is the
anti-derivative (or the indefinite integral) of f(u)
Where: C = integration constant, f(u) = integrnad,
F(u) = particular integral
EXAMPLE
adu au C af u du a f u du
faxdx afudu u ax dv fuvdu du fuvdv
uv dx udx vdx
udv uv vdu
fgduf gf g f
nnnn
=+ =
== =
±±… = ± ±
=−
=−
+
∫∫
∫∫∫ ∫
∫∫
∫∫
−− −
() ()
() (), (,) (,)
()
() ( ) ( ) (
1
12 3
(integration by parts)
))()()
()
,
ln
ln ,,
sin cos cos sin
sec tan csc cot
′′ −…+ −
=++≠
=+ =+
=+>
=− + = +
=+ =
∫∫
∫∫
∫∫
+
gfgdu
udu nuCn
du
uuC edue C
udu u C udu u C
udu u C udu u
nn
nn
uu
uu
1
1
11
101
1
22
(generalized integration by parts)
++
=+ =−+
=+ =+
=++ =−+
=+
+=+
=
∫∫
∫∫
∫∫
C
uudu uC uudu uC
udu u C udu u C
udu u u C udu u u C
du
au
u
aCa
du
ua a
u
aCa
du
uu a
sec tan sec csc cot csc
tan lnsec cot lnsin
sec lnsec tan csc lncsc cot
sin ,
tan ,
22
1
22
1
22
0
10
11 01
2
1
2
1
22
22
a
u
aCa du
au a
ua
ua C
du
ua a
ua
ua C
sec , ln
ln
+≠ =+
+
=
++
∫∫
uua auuaC
ua
udu u a a aua
uC
uuaua auuaC
22 22 222
22 2232
222222
222 2222 422
22
3
828
±= ±± +±+
±=
±+
+=+++
+
±= ± ± +±+
ln
()
ln
() ln
/
uua
udu u a a u
aCdu
ua
uuaC
ua
udu ua
uuuaC
udu
ua
uua auuaC
du
uu a a
u
22 22 1
22
22
22
2
22 22
2
22
22 222
22
2
22
1
=− + ±=+ ±+
±=− ±++ ±+
±=± +±+
+=− +
∫∫
sec ln
ln
ln
ln
aaa
uCdu
uu a a
u
aC
du
uu a
ua
au Cdu
ua
u
au a
C
2
22
1
22 2
22
22232
22 2
1++=+
±=±+±±+
∫∫
sec
()
/
audu
uau au
aC
ua udu au C
ua uduuuaau au
aC
au
udu a u a aau
uC
au
22 22 21
22 2232
22 2 2 2 2 2 41
22 22 22
22
22
3
828
−= −+ +
−=
+
−= − + +
=−+−
+
sin
()
() sin
ln
/
uu du uau u
aCudu
au
auC
udu
au
uau au
aC
du
ua u a
aau
uC
du
ua u au auC
du
a
2
22 1
22
22
2
22
22 21
22
22
22 2 2
22 2
1
22
1
1
∫∫
=− − − + =− − +
=− − + +
=− +−
+
=− − +
sin
sin
ln
(=+
−=+ +
u
u
aa u
C
au du uuaau au
aC
232 22 2
2232 2 222 41
825 3
8
)
() ( ) sin
/
/
Given:
Then:
W
d
du Fu C fu
Fu C fudu
d
[() ] ()
()
(
+= =
==
+
derivative
[ ( ) + ] anti-derivative or indefinite integral
d
Given:
Then:
d
d
du uC fu u
uC udu
C
()()
(
+= = =
==
derivative
(+) anti
-derivative or indefinite integral
4
43
3
4
4
CALCULUS INTEGRALS 1 A-798-61© 1996-2012 Mindsource Technologies Inc.
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Description
This basic quick reference study guide provides a clear summary of essential Integrals information, including a basic table and references to specialized Integrals such as trigonometric and hyperbolic integrals.
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