Calculus Cheat Sheet Integrals ...

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Calculus Cheat Sheet
Calculus Cheat Sheet
Integrals
Definitions
Standard Integration Techniques
Note that at many schools all but the Substitution Rule tend to be taught in a Calculus II class.
Definite Integral:
Suppose
fx
is continuous
Anti-Derivative :
An anti-derivative of
fx
b
gb
u
Substitution :
The substitution
ugx
$
will convert
fgxgxdx
)
$
fudu
using
on
'
ab
. Divide
'
(
(
)
ab
into
n
subintervals of
is a function,
Fx
, such that
Fxfx
$
.
a
ga
*
x
from each interval.
width
&
x
and choose
8
Indefinite Integral :
fxdxFxc
$
!
dugxdx
$
)
. For indefinite integrals drop the limits of integration.
+
b
a
*
2
2
8
where
Fx
is an anti-derivative of
fx
.
2
8
8
2
3
2
3
5
3
Then
fxdx
$
lim
fxx
&
.
Ex.
5cos
xdx
5cos
xdx
$
cos
udu
i
n
1
1
1
,+
i
$
1
3
2
2
8
ux
$1$
duxdx
3
1$
xdxdu
1
3
5
5
$
sin
u
$
sin8sin1
"
3
3
1
Fundamental Theorem of Calculus
$1$$
1
u
11::
3
$1$$
2
u
28
3
fx
is continuous on
'
(
Variants of Part I :
Part I :
If
ab
then
d
ux
a
x
a
8
)
ftdtuxfux
$
b
b
is also continuous on
'
(
b
a
$
8
gx
ftdt
ab
Integration by Parts :
udvuvvdu
$"
and
udvuv
$
"
vdu
. Choose
u
and
dv
from
dx
a
a
d
b
vx
d
$
8
x
a
integral and compute
du
by differentiating
u
and
com
pute
v
using
vdv
.
ftdt
$"
vxfvx
)
and
gx
)
$
ftdt
$
fx
.
dx
dx
5
3
ln
xdx
8
e
xdx
Ex.
"
d
8
fx
is continuous on
'
(
ux
vx
Ex.
Part II :
ab
,
Fx
is
'
(
'
(
ftdtuxf
$
"
vxf
ux
()
vx
()
dx
uxdv
$
$
"
1$
dudxv
$"
"
$
8
1
an anti-derivative of
fx
(
i.e.
Fx
fxdx
)
uxdvdx
$
ln
$1$
dudxvx
$
x
"
"
"
"
"
xdxx
$"!
dxx
$""!
ee
5
5
5
b
a
fxdxFbFa
5
3
ln
xdxxx
$
ln
"
dxxxx
$
ln
"
then
$
"
.
3
3
3
$
5ln53ln32
"
"
Properties
fxgxdx
-
$
fxdxgxdx
-
cfxdxcfxdx
$
,
c
is a constant
Products and (some) Quotients of Trig Functions
For sinco
n
x xd
8
we have the following :
1.
n
odd.
Strip 1 sine out and convert rest to
cosines using
For tanse
n
x xd
8
we have the following :
1.
n
odd.
Strip 1 tangent and 1 secant out and
convert the rest to secants using
2
b
b
b
b
b
fxgxdx
-
$
fxdx
-
gxdx
cfxdxcfxdx
$
,
c
is a constant
a
a
a
a
a
a
a
fxdx
$
b
b
8
0
fxdx
$
ftdt
sin1cos
2
$"
2
, then use
a
a
the substitution co
u
$ .
2.
m
odd.
Strip 1 cosine out and convert rest
to sines using
tan
$
sec
2
"
1
, then use the substitution
b
a
b
b
fxdx
$"
fxdx
se
u
$ .
2.
m
even.
Strip 2 secants out and convert rest
to tangents using
fxdx
*
fxdx
a
b
a
a
2
2
cos
$"
1sin
, then use
b
a
If
fxgx
.
on
axb
**
then
fxdx
.
gxdx
2
2
the substitution
si
u
$
.
3.
n
and
m
both odd.
Use either 1. or 2.
4.
n
and
m
both even.
Use double angle
and/or half angle formulas to reduce the
integral into a form that can be integrated.
sec
$!
1tan
, then
a
b
use the substitution ta
u
$ .
3.
n
odd and
m
even.
Use either 1. or 2.
4.
n
even and
m
odd.
Each integral will be
dealt with differently.
b
a
8
If
fx
. on
axb
0
**
then
fxdx
.
0
b
a
If
mfxM
*
*
on
axb
**
then
mba
"*
fxdxMba
*
"
2
2
Trig Formulas
:
sin22sincos
$
,
cos
$!
1
2
1cos2
,
sin
$"
1
2
1cos2
Common Integrals
5
3
8
8
8
8
3
5
sin
cos
x
x
dx
kdxkxc
$!
cos
udu
$
sin
uc
!
tan
udu
$
lnsec
uc
!
Ex.
tansec
xdx
8
Ex.
8
n
1
n
!
1
8
8
xdx
$
!/"
cn
1
sin
udu
$"
cos
uc
!
sec
udu
$
lnsectan
u
!
uc
!
tansec
3
5
xdx
$
tansectansec
2
4
xxxdx
22
5
4
(sin)
x
sin
x
sin
sinsin
xx
dx
$
dx
$
dx
n
!
1
3
3
3
cos
cos
cos
u
"
1
8
2
"
1
xdx
$
1
dx
$!
ln
xc
sec
udu
$
tan
uc
!
au
du
1
$
1
tan
!
2
4
$
sec
"
1sectansec
xxxdx
22
3
22
x
a
a
(1cos)
cos
(1)
"
sin
22
$
dx
u
$
cos
!
8
8
sectan
uudu
$
sec
uc
!
1
u
a
axb
dx
1
$
1
ln
axbc
!!
8
"
1
du
$
sin
!
2
4
$
u
"
1
udu
u
$
sec
a
!
"
u
24
22
12
"!
uu
au
"
$"
du
$"
du
8
csccot
uudu
$"!
csc
uc
3
3
8
u
u
ln
uduuuuc
$
ln
"!
$
1
sec
7
"
1
sec
5
xc
!
7
5
$
sec
2
!
2lncos
"
cos
2
xc
!
1
1
8
csc
2
udu
$"!
cot
uc
2
2
8
e
u
u
du
$!
©
2005 Paul Dawkins
©
2005 Paul Dawkins
Visit
for a complete set of Calculus notes.
Visit
for a complete set of Calculus notes.
 Calculus Cheat Sheet
Calculus Cheat Sheet
Trig Substitutions :
If the integral contains the following root use the given substitution and
for
mula to co
nvert into an integral inv
olving tri
g functions.
2
Applications of Integrals
b
a
fxdx
Net Area :
fx
and the
x
-axis with area above
x
-axis positive and area below
x
-axis negative.
represents the net area between
a
b
a
b
a
b
abx
"
22
1$
sin
bxa
22
"1$
2
sec
abx
2
!
22
1$
tan
2
2
2
2
2
2
cos
$"
1sin
tan
$
sec
"
1
sec
$!
1tan
16
49
Ex.
x
dx
9
:
16
12
sin
2
3
cos
d
$
d
Area Between Curves :
The general formulas for the two main cases for each are,
2
x
2
2
2
"
9
sin2cos
b
a
d
c
$
2
sin
1$
dx
2
cos
d
yfx
$
1$
A
"
dx
&
xfy
$
1$
A
"
dy
2
3
3
$
12csc
d
$"
12cot
!
upper function
lower function
right function
left function
$" $
"
Recall
2
x
$ . Because we have an indefinite
integral we’ll assume positive and drop absolute
value bars. If we had a definite integral we’d
need to compute
’s and remove absolute value
bars based on that and,
44sin
2
4cos
2
2cos
49
x
2
Use Right Triangle Trig to go back to
x
’s. From
substitution we have
If the curves intersect then the area of each portion must be found individually. Here are some
sketches of a couple possible situations and formulas for a couple of possible cases.
3
2
x
sin
$
so,
if
.
0
2
49
3
"
x
From this we see that
cot
$
. So,
$
x
"
if
#
0
2
16
449
"
x
dx
$"
!
d
c
In this case we have
2
$
2cos
.
b
a
49
x
"
b
x
Afygydy
$
"
2
2
x
49
"
x
Afxgxdx
$
"
Afxgxdx
$
"
!
gxfxdx
"
a
Px
Qx
dx
Partial Fractions :
If integrating
where the degree of
Px
is smaller than the degree of
Volumes of Revolution :
The two main formulas are
VAxdx
$
and
VAydy
$
. Here is
Qx
. Factor denominator as completely as possible and find the partial fraction decomposition of
the rational expression. Integrate the partial fraction decomposition (P.F.D.). For each factor in the
denominator we get term(s) in the decomposition according to the following table.
some general information about each method of computing and some examples.
Rings
Cylinders
2
2
A
$
"
A
$
2
outer radius
inner radius
radiuswidth / height
Limits:
x
/
y
of right/bot ring to
x
/
y
of left/top ring
Limits :
x
/
y
of inner cyl. to
x
/
y
of outer cyl.
Factor in
Qx
Term in P.F.D
Factor in
Qx
Term in P.F.D
Horz. Axis use
fx
,
Vert. Axis use
fy
,
Horz. Axis use
fy
,
Vert. Axis use
fx
,
A A
axb
axb
A
A
axb
1
!
2
!!
k
axb
!
axb
!
gx
,
Ax
and
dx
.
gy
,
Ay
and
dy
.
gy
,
Ay
and
dy
.
gx
,
Ax
and
dx
.
2
!
!
axb
!
!
AxB
axbxc
!
!!
!!
AxB
!
AxB
axbxc
!
!!
1
1
Ex.
Axis :
y
$%
0
Ex.
Axis :
y
$*
0
Ex.
Axis :
y
$%
0
Ex.
Axis :
y
$*
0
2
2
axbxc
axbxc
!!
!!
2
axbxc
2
!!
2
2
2
Ax
2
713
xx
x x
dx
"!
!
!
!
$! $
Set numerators equal and collect like terms.
A
BxC
!!
4)(
BxCx
!
)(
"
1
Ex.
8
713
2
2
x
"
1
2
2
1
)(
4
xx
"!
1
)(
4
!
4
xx
"!
1
)(
4
2
713
!
4
316
x
!
dx
$!
dx
2
x
"
1
2
xx
"!
1
4
x
x
!
4
)(
2
2
!$! !"!"
Set coefficients equal to get a system and solve
to get constants.
7
713
xABxCBxAC
4
$!!
4
3
16
dx
x
"
1
2
2
!
4
!
4
$
4ln1ln
"!
2
!!
48tan
"
1
3
2
radius :
ay
2
outer radius :
afx
"
outer radius:
agx
!
radius :
ay
!
AB
!$
CB
"$
134
AC
"$
0
"
Here is partial fraction form and recombined.
width :
fygy
"
A
$
4
B
$
3
C
$
16
inner radius :
agx
"
inner radius:
afx
!
width :
fygy
"
An alternate method that
sometimes
works to find constants. Start with setting numerators equal in
previous example :
These are only a few cases for horizontal axis of rotation. If axis of rotation is the
x
-axis use the
0
2
2
713
!$
xAx
!!!
4
BxCx
"
1
. Chose
nice
values of
x
and plug in.
y
$* case with
. For vertical axis of rotation (
x
$%
and
x
$*
) interchange
x
and
a
$
0
0
0
For example if
$
1
we get
205
A
$
which gives
A
$
4
. This won’t always work easily.
y
to get appropriate formulas.
©
2005 Paul Dawkins
©
2005 Paul Dawkins
Visit
for a complete set of Calculus notes.
Visit
for a complete set of Calculus notes.
 Calculus Cheat Sheet
Average Function Value :
The average value
of
Work :
If a force of
Fx
moves an object
1
b
8
b
a
fx
on
axb
**
is
$
fxdx
$
8
in
axb
**
, the work done is
WFxdx
avg
ba
"
a
Arc Length Surface Area :
Note that this is often a Calc II topic. The three basic formulas are,
b
Lds
b
a
b
a
$
8
(rotate about
y
-axis)
where
ds
is dep
endent upon the form of the function
being worked
with as follows.
$
8
$
8
SA
2
yds
(rotate about
x
-axis)
SA
2
xds
2
2
dy
dx
2
dy
dx
ds
$!
1
dxyfxaxb
if
$
**
ds
$
!
dt
if
xftygt
$
$
atb
**
dt
dt
2
2
dr
d
dx
dy
ds
$!
2
d
if
rf
$
a
**
b
ds
$!
1
dy
if
xfyayb
$
**
With surface area you
may
have to substitute in for the
x
or
y
depending on your choice of
ds
to
match the differential in the
ds
. With parametric and polar you will always need to substitute.
Improper Integral
An improper integral is an integral with one or more infinite limits and/or discontinuous integrands.
Integral is called convergent if the limit exists and has a finite value and divergent if the limit
doesn’t exist or has infinite value. This is typically a Calc II topic.
Infinite Limit
1.
+
t
b
b
fxdx
$
lim
fxdx
2.
fxdx
$
lim
fxdx
+
a
,+
a
"
,"+
+
c
+
3.
fxdx
$
fxdx
!
fxdx
provided BOTH integrals are convergent.
+
+
"
"
c
Discontinuous Integrand
1.
Discont. at
a
:
b
b
b
fxdx
$
lim
fxdx
2.
Discont. at
b
:
fxdx
$
lim
fxdx
a
ta
,
!
a
tb
,
"
a
b
b
3.
Discontinuity at
acb
##
:
fxdx
$
fxdx
!
fxdx
provided both are convergent.
a
a
on
'
Comparison Test for Improper Integrals :
If
fxgx
.
.
0
a
+ then,
+
+
+
+
8
8
8
8
1.
If
a
fxdx
conv. then
a
gxdx
conv.
2.
If
a
gxdx
divg. then
a
fxdx
divg.
+
8
1
Useful fact : If
a
%
0
then
dx
converges if
p
%
1
and diverges for
p
*
1
.
p
a
x
Approximating Definite Integrals
b
a
fxdx
ba
n
"
8
For given integral
and a
n
(must be even for Simpson’s Rule) define
&$
x
and
divide
'
ab
into
n
subintervals
'
(
xx
,
'
(
xx
, … ,
'
(
(
n
xx
1
,
with
0
xa
$
and
n
xb
$
then,
0
1
"
n
b
x
is midpoint
'
(
Midpoint Rule :
fxdxxfx
0&
*
!
fx
*
!!
fx
*
,
*
i
xx
1
,
"
i
1
2
n
a
&
b
8
Trapezoid Rule :
fxdx
0
fx
!
2
fx
!!
2
fx
!!
2
fx
!
fx
0
1
2
n
"
1
n
2
a
&
b
Simpson’s Rule :
fxdx
0
fx
!
4
fx
!
2
fx
!!
2
fx
!
4
fx
!
fx
0
1
2
n
"
2
n
"
1
n
3
a
©
2005 Paul Dawkins
Visit
for a complete set of Calculus notes.
  [ Pobierz całość w formacie PDF ]