Calculus Summary
Calculus Summary
Calculus Summary
com/notes 6/11/2001
CALCULUS SUMMARY
A quick reference on Exponents, Logarithms, Differentiation, Integration, Power Series
Exponents
b
x
> 0 b b
x y
if and only if x y
b b b
x y x y
+
( ) b b
x y xy
b b b
x y x y
/
a e
x x a
ln
Logarithms
Natural Logarithmic Function f x x x
e
( ) log ln The
natural number e 2 71828182846 . . To get this
number on the calculator, press 1 INV lnx.
log
e
x is written ln x (read "el - en - ex")
1/
0
lim(1 )
x
x
e x
+ ln x b if and only if e x
b
lim ln
x
x
+
0
limln
x
x
.
lne x
x
e b
a b a ln
ln ln ln xy x y + ln ln ln
x
y
x y
ln ln x y x
y
Logarithms to other bases:
y x
a
log if and only if
y
a x
log log log
a a a
xy x y +
log log log
a a a
x
y
x y
log log
a
y
a
x y x
log
log
log
a
b
b
x
x
a
A calculator can be used to evaluate an expression such as
log
2
14 by virtue of the fact that it is equivalent to ln14/ ln 2 .
RULES OF DIFFERENTIATION
(where u is a function of x)
The derivative of a constant is 0.
The power rule: the derivative of
n
x is
1 n
nx
.
The general power rule*:
d
dx
n n
u nu u
1
The constant multiple rule:
d
dx
c u c u ( )
The sum and difference rule:
d
dx
u v u v ( ) t t
*The General Power rule is a special case of the Chain rule.
The product rule:
d
dx
u v u v u v ( ) +
The quotient rule:
d
dx
u
v
v u u v
v
_
,
2
The chain rule*: ( ) ( )
d d d
dx du dx
f u f u u
The absolute value rule: , 0
d
dx
u
u u u
u
Exponential functions: ( ) ( )
d
dx
x x
a a a ln
( ) ( )
d
dx
u u
a a a u ln
The natural number e:
d
dx
u u
e e u ( )
The natural log:
d
dx
x
x
(ln )
1
d
dx
u
u
u
(ln )
d
dx
u x
u
x
x u ( ln ) ln +
Logarithms to other bases: ( )
d
dx a
x
a x
log
(ln )
1
( )
d
dx a
u
a u
u log
(ln )
1
Trigonometric formulas:
(sin ) cos
d
u u u
dx
(cos ) sin
d
u u u
dx
( )
2
tan sec
d
u u u
dx
(sec ) sec tan
d
u u u u
dx
2
(cot ) csc
d
u u u
dx
(csc ) csc cot
d
u u u u
dx
2
[arcsin ]
1
d u
u
dx
u
d
dx
u
u
u
[arccos ]
1
2
2
[arctan ]
1
d u
u
dx u
+
2
[arccot ]
1
d u
u
dx u
+
2
[arcsec ]
1
d u
u
dx
u u
2
[arccsc ]
1
d u
u
dx
u u
( ) sinh cosh
d
u u u
dx
( ) cosh sinh
d
u u u
dx
Example of the general power rule with trigonometric functions:
d
dx
x
d
dx
x x x x x (sin ) (sin ) (sin ) cos sin cos
3 3 2 2
3 3
Tom Penick tom@tomzap.com www.teicontrols.com/notes 6/11/2001
RULES OF INTEGRATION
The basic formula: x dx
x
n
C
n
n
+
+
+
1
1
Constants: 0 dx C
dx x c +
k dx kx C +
k f x dx k f x dx ( ) ( )
The sum and difference rule:
[ ( ) ( )] ( ) ( ) f x g x dx f x dx g x dx t t
Fractional functions:
1
x
dx x C +
ln
1
u
du u C +
ln
u
u
dx u C ln
Exponential functions: a dx
a
a C
x x
_
,
+
1
ln
The natural number e: e dx e C
x x
+
e dx
u
e C
u u
1
xe dx x e C
x x
+
( ) 1
2
( 1)
ax
ax
e
xe dx ax C
a
+
( )
( )
( )
( ) ( )
2
1 / 2
0
/ 2 1 / 2
1 / 2 !
for odd
2
1 3 5 1
for even
2
n
n ax
n n
n
n
a
x e dx
n
n
a
a
+
+
1
]
L
Composite function where u is a function of x:
f u u dx F u C ( ) ( ) +
The general power rule: u u dx
u
n
C
n
n
+
+
+
1
1
Integration by parts; try letting dv be the most
complicated portion of the
integrand that fits an
integration formula:
The definite integral where f(x) is the derivative of F(x):
[ ] f x dx F b F a F x
a
b
a
b
( ) ( ) ( ) ( )
Second degree polynomials for p x Ax Bx C ( ) + +
2
:
p x dx
b a
p a p
a b
p b
a
b
( ) ( ) ( )
_
,
+
+
_
,
+
1
]
1
6
4
2
Trigonometric formulas:
1
sin cos u dx u C
u
+
1
cos sin u dx u C
u
+
sec lnsec tan udu u u C + +
sec tan
2
udu u C +
sec tan sec u udu u C +
csc ln csc cot u du u u C + +
csc cot
2
udu u C +
csc cot csc u udu u C +
tan ln cos udu u C +
cot ln sin udu u C +
du
a u
u
a
C
2 2
arcsin
du
a u a
u
a
C
2 2
1
+
+
arctan
du
u u a
a
arc
u
a
C
2 2
1
sec
sin sin
2
1
2
1
4
2 udu u u C +
cos sin
2
1
2
1
4
2 udu u u C + +
sin ( sin ) cos
3 2
1
3
2 udu u u C + +
+ + C u u du u sin ) cos 2 (
3
1
cos
2 3
Definite Integrals
Natural Number e:
/
0
x
e dx
/ 2
0
x
xe dx
2 / 3
0
2
x
x e dx
/ 1
0
!
n x n
x e dx n
Probability Integrals
For the form
2
0
n ax
n
I x e dx
1/ 2
0
2
I a
1
1
2
I
a
3/ 2
2
4
I a
2
3
1
2
I a
5/ 2
4
3
8
I a
3
3
I a
for odd n:
( )
( ) 1 / 2
1 / 2 !
2
n
n
n
I
a
+
1
]
for even n:
( )
( ) ( ) / 2 1 / 2
1 3 5 1
2
n
n n
n
I
a a
+
L
Complex Trigonometric Identities
) ( cos
2
1
+
j j
e e +
sin cos j e
j
) ( sin
2
1
j j
j
e e
udv uv v du
Tom Penick tom@tomzap.com www.teicontrols.com/notes 6/11/2001
Quadratic Equation
Given the equation
0
2
+ + c bx ax : x
b b ac
a
t
2
4
2
Power Series Representation
...
! 3 ! 2
1
!
3 2
0
+ + + +
x x
x
n
x
e
n
n
x
cos
( )
( )! ! !
... x
x
n
x x
n n
n
1
2
1
2 4
2
0
2 4
sin
( )
( )! ! !
... x
x
n
x
x x
n n
n
+
+
+
1
2 1 3 5
2 1
0
3 5
cosh
( )! ! !
... x
x
n
x x
n
n
+ + +
2
0
2 4
2
1
2 4
sinh
( )! ! !
... x
x
n
x
x x
n
n
+
+ + +
+
2 1
0
3 5
2 1 3 5
ln( )
( )
... 1
1
2 3
1
0
2 3
+
+
+
x
x
n
x
x x
n n
n
... 1
1
1
3 2
0
+ + + +
x x x x
x
n
n
2 2 4 6
2
0
1
1 ..., 1
1
n
n
x x x x x
x
+ + + + <
1
1 , 1
1
x x
x
+
=
1
1 1 , 1
2
x x x + + =