august worksheet for precalculus and calculusbmaprecalculus.net/handouts/augustassignment.pdf ·...
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Name:___________________________
August Worksheet for PreCalculus and Calculus
The goal of this assignment is to give you experience in using the BMA Math Reference (attached to this worksheet). This is a very important document that I want all of you to use often in my classes.
Every question on this worksheet can be answered by finding the relevant information in the BMA Math Reference and then maybe doing a very simple mental calculation.
Please do not use a calculator or other external resources. These will detract from the goal of gaining familiarity with this document. (You are welcome to work with other students. Calculus students are especially encouraged to check in with PreCalculus students.)
1) What is the Complex Modulus of 3 + 2i?
2) What is the Complex Conjugate of 3 + 2i?
3) Solve for x: 2=log3 x
4) Solve for x: 5=12log12 x
5) Factor the following special polynomial: y=x3−6x2+12x−8
6) Factor completely: y=x16−216
page 1 of 4
7) Write out the quadratic formula.
8) Choose real numbers for a, b, and c so that y=ax2+bx+c has two real roots. (Give the roots.)
9) Choose real numbers for a, b, and c so that y=ax2+bx+c has one repeated root. (Give the root.)
10) Choose real numbers for a, b, and c so that y=ax2+bx+c has two imaginary roots. (Give the root.)
11) Pick real numbers for two points (a,b) and (c,d). Write out the equation of the line that contains those two points in point-slope form. Transform that equation into point-slope form.
12) What type of graph is associated with the equation: 1=( x−5)2
9−
( y+2)2
16
13) Choose your favorite "Common Algebraic Error." Write a few sentences explaining why this is your favorite error. Promise that you'll never make this error again.
page 2 of 4
14) An annulus has an outside radius of 6 and an inside radius of 4. What is the area of the annulus?
15) What is the surface area of a square pyramid with a base of 5 and a height of 2?
16) Given tan x=3 find sec x .
17) Given tan x=.3 and tan y=.7 find tan (x+ y ) .
18) Convert 330° into radians, and find sin (330 °) and cos(330 °) .
19) Fill in the blanks: The sine of 60° is also equal to the sine of _____°, the cosine of _____° and also the cosine of _____°.
20) A triangle has sides A, B, and C, and angles a, b, c. If A=12, sin (a)=0.3 and sin (b)=0.5 , find the length of side B.
page 3 of 4
21) Sketch the graph of y=x4−x2
22) Sketch the graph of y=log x
23) Solve for y: y=∫sin x dx
Bonus) Given that sin (36.87)=35
and sin (22.62)= 513
Find the following (without a calculator):
cos(36.87) cos(22.62)
sin (53.13) cos(53.13)
sin (67.38) cos(67.38)
sin (59.49) cos(59.49)
sin (14.25) cos(14.25)
page 4 of 4
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul Dawkins
Algebra Cheat Sheet
Basic Properties & Facts Arithmetic Operations
( )
, 0
b abab ac a b c a
c c
a
a a acb
bc bc b
c
a c ad bc a c ad bc
b d bd b d bd
a b b a a b a b
c d d c c c c
a
ab ac adbb c a
ca bc
d
! "+ = + =# $
% &
! "# $% & = =
! "# $% &
+ '+ = ' =
' ' += = +
' '
! "# $+ % &= + ( =! "# $% &
Exponent Properties
( )
( )
( ) ( )1
1
0
1
1, 0
1 1
nm
m m
nn m n m n m
m m n
mn nm
n nn n n
n
n n
n n
n n n nn
n
aa a a a
a a
a a a a
a aab a b
b b
a aa a
a b ba a a
b a a
+ '
'
'
'
'
= = =
= = (
! "= =# $
% &
= =
! " ! "= = = =# $ # $
% & % &
Properties of Radicals
1
, if is odd
, if is even
nn n n n
nm n nm n
n
n n
n n
a a ab a b
a aa a
b b
a a n
a a n
= =
= =
=
=
Properties of Inequalities If then and
If and 0 then and
If and 0 then and
a b a c b c a c b c
a ba b c ac bc
c c
a ba b c ac bc
c c
< + < + ' < '
< > < <
< < > >
Properties of Absolute Value
if 0
if 0
a aa
a a
)*= +
' <,
0
Triangle Inequality
a a a
aaab a b
b b
a b a b
) ' =
= =
+ - +
Distance Formula
If ( )1 1 1,P x y= and ( )2 2 2,P x y= are two
points the distance between them is
( ) ( ) ( )2 2
1 2 2 1 2 1,d P P x x y y= ' + '
Complex Numbers
( ) ( ) ( )
( ) ( ) ( )
( )( ) ( )
( )( )
( )
( )( )
2
2 2
2 2
2
1 1 , 0
Complex Modulus
Complex Conjugate
i i a i a a
a bi c di a c b d i
a bi c di a c b d i
a bi c di ac bd ad bc i
a bi a bi a b
a bi a b
a bi a bi
a bi a bi a bi
= ' = ' ' = )
+ + + = + + +
+ ' + = ' + '
+ + = ' + +
+ ' = +
+ = +
+ = '
+ + = +
Burke Mountain AcademyMath Reference
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul Dawkins
Logarithms and Log Properties Definition
log is equivalent to y
by x x b= =
Example 3
5log 125 3 because 5 125= =
Special Logarithms
10
ln log natural log
log log common log
ex x
x x
=
=
where 2.718281828e = K
Logarithm Properties
( )( )
log
log 1 log 1 0
log
log log
log log log
log log log
b
b b
xx
b
r
b b
b b b
b b b
b
b x b x
x r x
xy x y
xx y
y
= =
= =
=
= +
! "= #$ %
& '
The domain of logb
x is 0x >
Factoring and Solving Factoring Formulas
( ) ( )
( )
( )
( ) ( ) ( )
2 2
22 2
22 2
2
2
2
x a x a x a
x ax a x a
x ax a x a
x a b x ab x a x b
# = + #
+ + = +
# + = #
+ + + = + +
( )
( )
( )( )
( ) ( )
33 2 2 3
33 2 2 3
3 3 2 2
3 3 2 2
3 3
3 3
x ax a x a x a
x ax a x a x a
x a x a x ax a
x a x a x ax a
+ + + = +
# + # = #
+ = + # +
# = # + +
( )( )2 2n n n n n nx a x a x a# = # +
If n is odd then,
( ) ( )
( )( )
1 2 1
1 2 2 3 1
n n n n n
n n
n n n n
x a x a x ax a
x a
x a x ax a x a
# # #
# # # #
# = # + + +
+
= + # + # +
L
L
Quadratic Formula
Solve 2 0ax bx c+ + = , 0a ( 2 4
2
b b acx
a
# ± #=
If 2 4 0b ac# > - Two real unequal solns.
If 2 4 0b ac# = - Repeated real solution.
If 2 4 0b ac# < - Two complex solutions.
Square Root Property
If 2x p= then x p= ±
Absolute Value Equations/Inequalities
If b is a positive number
or
or
p b p b p b
p b b p b
p b p b p b
= ) = # =
< ) # < <
> ) < # >
Completing the Square
Solve 22 6 10 0x x# # =
(1) Divide by the coefficient of the 2x
2 3 5 0x x# # =
(2) Move the constant to the other side. 2 3 5x x# =
(3) Take half the coefficient of x, square
it and add it to both sides 2 2
2 3 3 9 293 5 5
2 2 4 4x x
! " ! "# + # = + # = + =$ % $ %
& ' & '
(4) Factor the left side 2
3 29
2 4x! "# =$ %
& '
(5) Use Square Root Property
3 29 29
2 4 2x # = ± = ±
(6) Solve for x
3 29
2 2x = ±
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul Dawkins
Functions and Graphs Constant Function
( )or y a f x a= =
Graph is a horizontal line passing
through the point ( )0,a .
Line/Linear Function
( )or y mx b f x mx b= + = +
Graph is a line with point ( )0,b and
slope m.
Slope
Slope of the line containing the two
points ( )1 1,x y and ( )2 2,x y is
2 1
2 1
rise
run
y ym
x x
!= =
!
Slope – intercept form
The equation of the line with slope m
and y-intercept ( )0,b is
y mx b= +
Point – Slope form
The equation of the line with slope m
and passing through the point ( )1 1,x y is
( )1 1y y m x x= + !
Parabola/Quadratic Function
( ) ( ) ( )2 2
y a x h k f x a x h k= ! + = ! +
The graph is a parabola that opens up if
0a > or down if 0a < and has a vertex
at ( ),h k .
Parabola/Quadratic Function
( )2 2y ax bx c f x ax bx c= + + = + +
The graph is a parabola that opens up if
0a > or down if 0a < and has a vertex
at ,2 2
b bf
a a
" #" #! !$ %$ %
& '& '.
Parabola/Quadratic Function
( )2 2x ay by c g y ay by c= + + = + +
The graph is a parabola that opens right
if 0a > or left if 0a < and has a vertex
at ,2 2
b bg
a a
" #" #! !$ %$ %& '& '
.
Circle
( ) ( )2 2 2
x h y k r! + ! =
Graph is a circle with radius r and center
( ),h k .
Ellipse
( ) ( )2 2
2 21
x h y k
a b
! !+ =
Graph is an ellipse with center ( ),h k
with vertices a units right/left from the
center and vertices b units up/down from
the center.
Hyperbola
( ) ( )2 2
2 21
x h y k
a b
! !! =
Graph is a hyperbola that opens left and
right, has a center at ( ),h k , vertices a
units left/right of center and asymptotes
that pass through center with slope b
a± .
Hyperbola
( ) ( )2 2
2 21
y k x h
b a
! !! =
Graph is a hyperbola that opens up and
down, has a center at ( ),h k , vertices b
units up/down from the center and
asymptotes that pass through center with
slope b
a± .
For a complete set of online Algebra notes visit http://tutorial.math.lamar.edu. © 2005 Paul Dawkins
Common Algebraic Errors Error Reason/Correct/Justification/Example
20
0! and
22
0! Division by zero is undefined!
23 9" ! 23 9" = " , ( )
23 9" = Watch parenthesis!
( )3
2 5x x! ( )
32 2 2 2 6x x x x x= =
a a a
b c b c! +
+
1 1 1 12
2 1 1 1 1= ! +=
+
2 3
2 3
1x x
x x
" "! ++
A more complex version of the previous
error.
a bx
a
+1 bx! +
1a bx a bx bx
a a a a
+= + = +
Beware of incorrect canceling!
( )1a x ax a" " ! " " ( )1a x ax a" " = " +
Make sure you distribute the “-“!
( )2 2 2
x a x a+ ! + ( ) ( ) ( )2 2 22x a x a x a x ax a+ = + + = + +
2 2x a x a+ ! + 2 2 2 25 25 3 4 3 4 3 4 7= = + ! + = + =
x a x a+ ! + See previous error.
( )n n n
x a x a+ ! + and n n nx a x a+ ! +
More general versions of previous three
errors.
( ) ( )2 2
2 1 2 2x x+ ! +
( ) ( )2 2 22 1 2 2 1 2 4 2x x x x x+ = + + = + +
( )2 22 2 4 8 4x x x+ = + +
Square first then distribute!
( ) ( )2 2
2 2 2 1x x+ ! +
See the previous example. You can not
factor out a constant if there is a power on
the parethesis!
2 2 2 2x a x a" + ! " + ( )
12 2 2 2 2x a x a" + = " +
Now see the previous error.
a ab
b c
c
!# $% &' (
1
1
a
a a c ac
b b b b
c c
# $% &
# $# $' (= = =% &% &# $ # $ ' (' (% & % &' ( ' (
a
acb
c b
# $% &' ( !
1
1
a a
a ab b
cc b c bc
# $ # $% & % &
# $# $' ( ' (= = =% &% &# $ ' (' (% &' (
2D GEOMETRY FORMULAS
SQUARE
s = sideArea: A = s2
Perimeter: P = 4s
s
s
RECTANGLE
l = length, w = widthArea: A = lw
Perimeter: P = 2l + 2w
w
l
TRIANGLE
b = base, h = heightArea: A = 1
2bh
Perimeter: P = a + b + c
h
b
a c
EQUILATERAL TRIANGLE
s = sideHeight: h =
!
3
2s
Area: A =!
3
4s2
h
s
s s
PARALLELOGRAM
b = base, h = height, a = sideArea: A = bh
Perimeter: P = 2a + 2b
b
h a
TRAPEZOID
a, b = bases; h = height;c, d = sidesArea: A = 1
2(a + b)h
Perimeter:P = a + b + c + d b
h
a
c d
CIRCLE
r = radius, d = diameterDiameter: d = 2rArea: A = !r2
Circumference: C = 2!r = !d r
d
SECTOR OF CIRCLE
r = radius, " = angle in radiansArea: A = 1
2"r2
Arc Length: s = "rr
s
"
ELLIPSE
a = semimajor axisb = semiminor axisArea: A = !ab
a
b
Circumference:C ! !
!
3(a + b) ""
(a + 3b)(b + 3a)#
ANNULUS
r = inner radius,R = outer radiusAverage Radius: # = 1
2(r + R)
Width: w = R " r
Area: A = !(R2 " r2)or A = 2!#w
r
R
REGULAR POLYGON
s = side length,n = number of sidesCircumradius: R = 1
2s csc(!
n)
Area: A = 1
4ns2 cot(!
n)
or A = 1
2nR2 sin(2!
n)
R
s
3D GEOMETRY FORMULAS
CUBE
s = sideVolume: V = s3
Surface Area: S = 6s2
ss
s
RECTANGULAR SOLID
l = length, w = width,h = heightVolume: V = lwh
Surface Area:S = 2lw + 2lh + 2wh
l
w
h
SPHERE
r = radiusVolume: V = 4
3!r3
Surface Area: S = 4!r2
r
RIGHT CIRCULAR
CYLINDER
r = radius, h = heightVolume: V = !r2h
Surface Area: S = 2!rh + 2!r2
r
h
TORUS
r = tube radius,R = torus radiusVolume: V = 2!2r2R
Surface Area: S = 4!2rR
R r
GENERAL CONE
OR PYRAMID
A = area of base, h = heightVolume: V = 1
3Ah
h
A
RIGHT CIRCULAR CONE
r = radius, h = heightVolume: V = 1
3!r2h
Surface Area:S = !r
!r2 + h2 + !r2
h
r
FRUSTUM OF A CONE
r = top radius, R = base radius,h = height, s = slant heightVolume: V = !
3(r2 + rR + R2)h
Surface Area:S = !s(R + r) + !r2 + !R2
s
h R
r
SQUARE PYRAMID
s = side, h = heightVolume: V = 1
3s2h
Surface Area:S = s(s +
!s2 + 4h2)
h
s
s
REGULAR TETRAHEDRON
s = sideVolume: V = 1
12
!2s3
Surface Area: S =!
3s2
s
© 2005 Paul Dawkins
Trig Cheat Sheet
Definition of the Trig Functions
Right triangle definition
For this definition we assume that
02
!"# # or 0 90"$ # # $ .
opposite
sinhypotenuse
" % hypotenuse
cscopposite
" %
adjacentcos
hypotenuse" %
hypotenusesec
adjacent" %
oppositetan
adjacent" %
adjacentcot
opposite" %
Unit circle definition
For this definition " is any angle.
sin1
yy" % %
1csc
y" %
cos1
xx" % %
1sec
x" %
tany
x" % cot
x
y" %
Facts and Properties Domain
The domain is all the values of " that
can be plugged into the function.
sin" , " can be any angle
cos" , " can be any angle
tan" , 1
, 0, 1, 2,2
n n" !& '( ) % * *+ ,- .
!
csc" , , 0, 1, 2,n n" !( % * * !
sec" , 1
, 0, 1, 2,2
n n" !& '( ) % * *+ ,- .
!
cot" , , 0, 1, 2,n n" !( % * * !
Range
The range is all possible values to get
out of the function.
1 sin 1"/ 0 0 csc 1 and csc 1" "1 0 /
1 cos 1"/ 0 0 sec 1 andsec 1" "1 0 /
tan"/2 0 0 2 cot"/2 0 0 2
Period
The period of a function is the number,
T, such that 3 4 3 4f T f" ") % . So, if 5
is a fixed number and " is any angle we
have the following periods.
3 4sin 5" 6 2
T!5
%
3 4cos 5" 6 2
T!5
%
3 4tan 5" 6 T!5
%
3 4csc 5" 6 2
T!5
%
3 4sec 5" 6 2
T!5
%
3 4cot 5" 6 T!5
%
"
adjacent
opposite
hypotenuse
x
y
3 4,x y
"
x
y 1
© 2005 Paul Dawkins
Formulas and Identities Tangent and Cotangent Identities
sin costan cot
cos sin
! !! !
! !" "
Reciprocal Identities
1 1csc sin
sin csc
1 1sec cos
cos sec
1 1cot tan
tan cot
! !! !
! !! !
! !! !
" "
" "
" "
Pythagorean Identities 2 2
2 2
2 2
sin cos 1
tan 1 sec
1 cot csc
! !
! !
! !
# "
# "
# "
Even/Odd Formulas
$ % $ %$ % $ %$ % $ %
sin sin csc csc
cos cos sec sec
tan tan cot cot
! ! ! !
! ! ! !
! ! ! !
& " & & " &
& " & "
& " & & " &
Periodic Formulas
If n is an integer.
$ % $ %$ % $ %$ % $ %
sin 2 sin csc 2 csc
cos 2 cos sec 2 sec
tan tan cot cot
n n
n n
n n
! ' ! ! ' !
! ' ! ! ' !
! ' ! ! ' !
# " # "
# " # "
# " # "
Double Angle Formulas
$ %$ %
$ %
2 2
2
2
2
sin 2 2sin cos
cos 2 cos sin
2cos 1
1 2sin
2 tantan 2
1 tan
! ! !
! ! !
!
!!
!!
"
" &
" &
" &
"&
Degrees to Radians Formulas
If x is an angle in degrees and t is an
angle in radians then
180and
180 180
t x tt x
x
' ''
" ( " "
Half Angle Formulas
$ %$ %
$ %$ %$ %$ %
2
2
2
1sin 1 cos 2
2
1cos 1 cos 2
2
1 cos 2tan
1 cos 2
! !
! !
!!
!
" &
" #
&"
#
Sum and Difference Formulas
$ %$ %
$ %
sin sin cos cos sin
cos cos cos sin sin
tan tantan
1 tan tan
) * ) * ) *
) * ) * ) *
) *) *
) *
+ " +
+ "
++ "
!
!
Product to Sum Formulas
$ % $ %
$ % $ %
$ % $ %
$ % $ %
1sin sin cos cos
2
1cos cos cos cos
2
1sin cos sin sin
2
1cos sin sin sin
2
) * ) * ) *
) * ) * ) *
) * ) * ) *
) * ) * ) *
" & & #, -. /
" & # #, -. /
" # # &, -. /
" # & &, -. /
Sum to Product Formulas
sin sin 2sin cos2 2
sin sin 2cos sin2 2
cos cos 2cos cos2 2
cos cos 2sin sin2 2
) * ) *) *
) * ) *) *
) * ) *) *
) * ) *) *
# &0 1 0 1# " 2 3 2 34 5 4 5
# &0 1 0 1& " 2 3 2 34 5 4 5
# &0 1 0 1# " 2 3 2 34 5 4 5
# &0 1 0 1& " & 2 3 2 34 5 4 5
Cofunction Formulas
sin cos cos sin2 2
csc sec sec csc2 2
tan cot cot tan2 2
' '! ! ! !
' '! ! ! !
' '! ! ! !
0 1 0 1& " & "2 3 2 34 5 4 5
0 1 0 1& " & "2 3 2 34 5 4 5
0 1 0 1& " & "2 3 2 34 5 4 5
© 2005 Paul Dawkins
Unit Circle
For any ordered pair on the unit circle ! ",x y : cos x# $ and sin y# $
Example
5 1 5 3cos sin
3 2 3 2
% %& ' & '$ $ () * ) *+ , + ,
3
%
4
%
6
%
2 2,
2 2
& ') *) *+ ,
3 1,
2 2
& ') *) *+ ,
1 3,
2 2
& ') *) *+ ,
60-
45-
30-
2
3
%
3
4
%
5
6
%
7
6
%
5
4
%
4
3
%
11
6
%
7
4
%
5
3
%
2
%
%
3
2
%
0
2%
1 3,
2 2
& '() *+ ,
2 2,
2 2
& '() *+ ,
3 1,
2 2
& '() *+ ,
3 1,
2 2
& '( () *+ ,
2 2,
2 2
& '( () *+ ,
1 3,
2 2
& '( () *+ ,
3 1,
2 2
& '() *
+ ,
2 2,
2 2
& '() *
+ ,
1 3,
2 2
& '() *
+ ,
! "0,1
! "0, 1(
! "1,0(
90-
120-
135-
150-
180-
210-
225-
240-270- 300-
315-
330-
360-
0-
x
! "1,0
y
© 2005 Paul Dawkins
Inverse Trig Functions Definition
1
1
1
sin is equivalent to sin
cos is equivalent to cos
tan is equivalent to tan
y x x y
y x x y
y x x y
!
!
!
" "
" "
" "
Domain and Range
Function Domain Range
1siny x!" 1 1x! # # 2 2
y$ $
! # #
1cosy x!" 1 1x! # # 0 y $# #
1tany x!" x!% & & % 2 2
y$ $
! & &
Inverse Properties
' (' ( ' (' (
' (' ( ' (' (
' (' ( ' (' (
1 1
1 1
1 1
cos cos cos cos
sin sin sin sin
tan tan tan tan
x x
x x
x x
) )
) )
) )
! !
! !
! !
" "
" "
" "
Alternate Notation 1
1
1
sin arcsin
cos arccos
tan arctan
x x
x x
x x
!
!
!
"
"
"
Law of Sines, Cosines and Tangents
Law of Sines
sin sin sin
a b c
* + ," "
Law of Cosines 2 2 2
2 2 2
2 2 2
2 cos
2 cos
2 cos
a b c bc
b a c ac
c a b ab
*
+
,
" - !
" - !
" - !
Mollweide’s Formula
' (12
12
cos
sin
a b
c
* +
,
!-"
Law of Tangents
' (
' (
' (
' (
' (
' (
12
12
12
12
12
12
tan
tan
tan
tan
tan
tan
a b
a b
b c
b c
a c
a c
* +
* +
+ ,
+ ,
* ,
* ,
!!"
- -
!!"
- -
!!"
- -
c a
b
*.
+.
,.
!
f x( ) = x0
= 1
!
f x( ) = x1
= x
!
f x( ) = x2
!
f x( ) = x1
2 = x
!
f x( ) = x3
!
f x( ) = x3" x
!
f x( ) = x4
!
f x( ) = x4" x
2
Di�erentiation Formulas
d
dxk = 0 (1)
d
dx[f(x) ± g(x)] = f ⇥(x) ± g⇥(x) (2)
d
dx[k · f(x)] = k · f ⇥(x) (3)
d
dx[f(x)g(x)] = f(x)g⇥(x) + g(x)f ⇥(x) (4)
d
dx
�f(x)g(x)
⇥=
g(x)f ⇥(x)� f(x)g⇥(x)[g(x)]2
(5)
d
dxf(g(x)) = f ⇥(g(x)) · g⇥(x) (6)
d
dxxn = nxn�1 (7)
d
dxsinx = cos x (8)
d
dxcos x = � sinx (9)
d
dxtanx = sec2 x (10)
d
dxcot x = � csc2 x (11)
d
dxsec x = sec x tanx (12)
d
dxcsc x = � csc x cot x (13)
d
dxex = ex (14)
d
dxax = ax ln a (15)
d
dxln |x| =
1x
(16)
d
dxsin�1 x =
1⇥1� x2
(17)
d
dxcos�1 x =
�1⇥1� x2
(18)
d
dxtan�1 x =
1x2 + 1
(19)
d
dxcot�1 x =
�1x2 + 1
(20)
d
dxsec�1 x =
1|x|⇥
x2 � 1(21)
d
dxcsc�1 x =
�1|x|⇥
x2 � 1(22)
Integration Formulas
⇤dx = x + C (1)
⇤xn dx =
xn+1
n + 1+ C (2)
⇤dx
x= ln |x| + C (3)
⇤ex dx = ex + C (4)
⇤ax dx =
1ln a
ax + C (5)
⇤lnx dx = x lnx� x + C (6)
⇤sinx dx = � cos x + C (7)
⇤cos x dx = sinx + C (8)
⇤tanx dx = � ln | cos x| + C (9)
⇤cot x dx = ln | sinx| + C (10)
⇤sec x dx = ln | sec x + tanx| + C (11)
⇤csc x dx = � ln | csc x + cot x| + C (12)
⇤sec2 x dx = tan x + C (13)
⇤csc2 x dx = � cot x + C (14)
⇤sec x tanx dx = sec x + C (15)
⇤csc x cot x dx = � csc x + C (16)
⇤dx⇥
a2 � x2= sin�1 x
a+ C (17)
⇤dx
a2 + x2=
1a
tan�1 x
a+ C (18)
⇤dx
x⇥
x2 � a2=
1a
sec�1 |x|a
+ C (19)
Standard Normal Distribution Table
0 z
z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
0.0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359
0.1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753
0.2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141
0.3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517
0.4 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879
0.5 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224
0.6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549
0.7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852
0.8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133
0.9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389
1.0 .3413 .3438 .3461 .3485 .3508 .3531 .3554 .3577 .3599 .3621
1.1 .3643 .3665 .3686 .3708 .3729 .3749 .3770 .3790 .3810 .3830
1.2 .3849 .3869 .3888 .3907 .3925 .3944 .3962 .3980 .3997 .4015
1.3 .4032 .4049 .4066 .4082 .4099 .4115 .4131 .4147 .4162 .4177
1.4 .4192 .4207 .4222 .4236 .4251 .4265 .4279 .4292 .4306 .4319
1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441
1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545
1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633
1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706
1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767
2.0 .4772 .4778 .4783 .4788 .4793 .4798 .4803 .4808 .4812 .4817
2.1 .4821 .4826 .4830 .4834 .4838 .4842 .4846 .4850 .4854 .4857
2.2 .4861 .4864 .4868 .4871 .4875 .4878 .4881 .4884 .4887 .4890
2.3 .4893 .4896 .4898 .4901 .4904 .4906 .4909 .4911 .4913 .4916
2.4 .4918 .4920 .4922 .4925 .4927 .4929 .4931 .4932 .4934 .4936
2.5 .4938 .4940 .4941 .4943 .4945 .4946 .4948 .4949 .4951 .4952
2.6 .4953 .4955 .4956 .4957 .4959 .4960 .4961 .4962 .4963 .4964
2.7 .4965 .4966 .4967 .4968 .4969 .4970 .4971 .4972 .4973 .4974
2.8 .4974 .4975 .4976 .4977 .4977 .4978 .4979 .4979 .4980 .4981
2.9 .4981 .4982 .4982 .4983 .4984 .4984 .4985 .4985 .4986 .4986
3.0 .4987 .4987 .4987 .4988 .4988 .4989 .4989 .4989 .4990 .4990
3.1 .4990 .4991 .4991 .4991 .4992 .4992 .4992 .4992 .4993 .4993
3.2 .4993 .4993 .4994 .4994 .4994 .4994 .4994 .4995 .4995 .4995
3.3 .4995 .4995 .4995 .4996 .4996 .4996 .4996 .4996 .4996 .4997
3.4 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4997 .4998
3.5 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .4998 .4998
Gilles Cazelais. Typeset with LATEX on April 20, 2006.