all maths formulas from maths.org
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Algebra Formulas
1. Set identities
Definitions:
I: Universal set
A: Complement
Empty set:
Union of sets
{ }|A B x x x BrA o =
Intersection of sets
{ }|A B x x x BdA an =
Complement
{ }|A x I x A =
Difference of sets
{ }\ |B A x x B x Aand=
Cartesian product
( ){ }, |A B x y x A and y B =
Set identities involving union
Commutativity
A B B A =
Associativity( ) ( )A B C A B C =
Idempotency
A A A =
Set identities involving intersection
commutativity
A B B A = Associativity
( ) ( )A B C A B C =
Idempotency
A A A =
Set identities involving union and intersection
Distributivity
( ) ( ) ( )A B C A B A C =
( ) ( ) ( )A B C A B A C =
Domination
A =
A I I =
Identity
A A =
A I A =
Set identities involving union, intersection and
complement
complement of intersection and union
A A I =
A A =
De Morgans laws
( )A B A B =
( )A B A B =
Set identities involving difference
( )\B A B A B=
\B A B A=
\A A =
( ) ( ) ( )\ \A B C A C B C =
\A I A =
2. Sets of Numbers
Definitions:N: Natural numbers
No: Whole numbers
Z: Integers
Z+: Positive integers
Z-: Negative integers
Q: Rational numbers
C: Complex numbers
Natural numbers (counting numbers )
{ }1, 2, 3,...N =
Whole numbers ( counting numbers + zero )
{ }0, 1, 2, 3,...oN =
Integers
{ }1, 2, 3,...Z N+ = =
{ }..., 3, 2, 1Z =
{ } { }0 . .., 3, 2, 1, 0, 1, 2, 3,...Z Z Z= =
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Irrational numbers:
Nonerepeating and nonterminating integers
Real numbers:
Union of rational and irrational numbers
Complex numbers:
{ }|C x iy x R and y R= +
N Z Q R C
3. Complex numbers
Definitions:
A complex nuber is written as a + bi where a and b arereal numbers an i, called the imaginary unit, has theproperty that i
2=-1.
The complex numbers a+bi and a-bi are called complex
conjugate of each other.
Equality of complex numbers
a + bi = c + di if and only if a = c and b = d
Addition of complex numbers
(a + bi) + (c + di) = (a + c) + (b + d)i
Subtraction of complex numbers
(a + bi) - (c + di) = (a - c) + (b - d)i
Multiplication of complex numbers
(a + bi)(c + di) = (ac - bd) + (ad + bc)i
Division of complex numbers
2 2 2 2
a bi a bi c di ac bd bc ad i
c di c di c di c d c d
+ + + = = +
+ + + +
Polar form of complex numbers
( )cos sin modulus, amplitudex iy r i r + = +
Multiplication and division in polar form
( ) ( )
( ) ( )
1 1 1 2 2 2
1 2 1 2 1 2
cos sin cos sin
cos sin
r i r i
r r i
+ + =
= + + +
( )
( )( ) ( )
1 1 1 11 2 1 2
2 2 2 2
cos sincos sin
cos sin
r r
r r
+ = +
+
De Moivres theorem
( ) ( )cos sin cos sinn nr r n n + = +
Roots of complex numbers
( )11 2 2
cos sin cos sinnnk k
r rn n
+ + + = +
From this the n nth roots can be obtained by putting k = 0,1, 2, . . ., n - 1
4. Factoring and product
Factoring Formulas
( )( )2 2a b a b a b = +
( )( )3 3 2 2a b a b a ab b = + +
( )( )3 3 2 2a b a b a ab b+ = + +
4 4 2 2( )( )( )a b a b a b a b = + +
( )( )5 5 4 3 2 2 3 4a b a b a a b a b ab b = + + + +
Product Formulas
2 2 2( ) 2a b a ab b+ = + +
2 2 2( ) 2a b a ab b = +
3 3 2 2 3( ) 3 3a b a a b ab b+ = + + +
3 3 2 2 3( ) 3 3a b a a b ab b = +
( )4 4 3 2 2 3 4
4 6 4a b a a b a b ab b+ = + + + +
( )4 4 3 2 2 3 44 6 4a b a a b a b ab b = + +
2 2 2 2( ) 2 2 2a b c a b c ab ac bc+ + = + + + + +
2 2 2 2( ...) ...2( ...)a b c a b c ab ac bc+ + + = + + + + + +
5. Algebric equations
Quadric Eqation: ax2
+ bx + c = 0
Solutions (roots):
2
1,2 42
b b acx
a
=
if D=b2-4ac is the discriminant, then the roots are
(i) real and unique if D > 0
(ii) real and equal if D = 0
(iii) complex conjugate if D < 0
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Cubic Eqation: 3 21 2 3 0x a x a x a+ + + =
Let
32
1 2 3 12 1
3 2 3 23 3
9 27 23,
9 54
,
a a a aa aQ R
S R Q R T R Q R
= =
= + + = +
then solutions are:
( ) ( )
( ) ( )
1 1
2 1
3 1
1
3
1 1 13
2 3 2
1 1 13
2 3 2
x S T a
x S T a i S T
x S T a i S T
= +
= + +
= +
if D = Q3
+ R3
is the discriminant, then:
(i) one root is real and two complex conjugate if D > 0
(ii) all roots are real and at last two are equal if D = 0(iii) all roots are real and unequal if D < 0
Cuadric Eqation: 24
4 31 2 3 0x ax a x a x a ++ + + =
Let y1 be a real root of the cubic equation
( ) ( )3 2 2 22 1 3 4 2 4 3 1 44 4 0y a y a a a y a a a a a + + = Solution are the 4 roots of
( ) ( )2 2 21 1 2 1 1 1 41 14 4 4 02 2
z a a a y z y y a+ + + =
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Functions Formulas
1. Exponents
... 0,p
p
a a a a p N p a Rif= >
0 1 0ia f a=
r s r s
a a a + = r
r s
s
aa
a
=
( )s
r r sa a
=
( )r r r
a b a b =
r r
r
a a
b b
=
1rr
aa
=
r
s rsa a=
2. Logarithms
Definition:
( )log , 0,yay x a x a x y R= = >
Formulas:
log 1 0a =
log 1a a =
log log loga a amn m n= +
log log loga a am
m nn
=
log logna am n m=
log log loga b am m b= log
loglog
ba
b
mm
a=
1og
loga
b
l ba
=
( )ln
og og lnln
a a
xl x l e x
a= =
3. Roots
Definitions:
a,b: bases ( , 0 2a b if n k = )
n,m: powers
Formulas:
n n nab a b=
nm m nn ma b a b=
, 0n
nn
a ab
b b=
, 0mn
nmnm
a ab
bb=
( )p
n nm mpa a=
( )n
n a a=
npn m mpa a=
m n mna a=
( )m
n mn a a=
1
1 , 0
n n
na aaa
=
2 2
2 2
a a b a a ba b
+ =
1 a b
a ba b=
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4. Trigonometry
Right-Triangle Definitions
Oppositesin
Hypotenuse
=
Adjacent
cosHypotenuse
=
Opposite
Adjacenttg =
1 Hypotenusecsc
sin Opposite
= =
1 Adjacentcot
Oppositetg
= =
1 Hypotenuseseccos Adjacent
= =
Reduction Formulas
sin( ) sinx x =
cos( ) cosx x =
sin( ) cos2
x x
=
cos( ) sin2 x x
=
sin( ) cos2
x x
+ =
cos( ) sin2
x x
=
sin( ) sinx x =
cos( ) cosx x =
sin( ) sinx x + =
cos( ) cosx x + =
Identities
2 2sin cos 1x x+ =
2
2
11
costg x
x+ =
2
2
1cot 1
sinx
x+ =
Sum and Difference Formulas
( )sin sin cos sin cos + = +
( )sin sin cos sin cos = ( )cos cos cos sin sin + =
( )cos cos cos sin sin = +
( )tan tan
tan1 tan tan
++ =
( )tan tan
tan1 tan tan
=
+
Double Angle and Half Angle Formulas
( )sin 2 2sin cos =
( ) 2 2cos 2 cos sin =
( )2
2tan 2
1
tg
tg
=
1 cossin
2 2
=
1 coscos
2 2
+=
1 cos sintan
2 sin 1 cos
= =
+
Other Useful Trig Formulae
Law of sines
sin sin sin
a b c
= =
Law of cosines
2 2 2
2 cosc a b ab = + Area of triangle
1sin
2K ab =
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5. Hyperbolic functions
Definitions:
sinh2
x xe e
x
=
cosh2
x xe ex+
=
sinhtanh
cosh
x x
x x
e e xx
e e x
= =
+
2 1csch
sinhx xx
e e x
= =
2 1sech
coshx x
xe e x
= =
+
coshcoth
sinh
x x
x x
e e xx
e e x
+= =
Derivates
sinh coshd
x xdx
=
cosh sinhd
x xdx
=
2tanh sechd
x xdx
=
csch csch cothd
x x xdx
=
sech sech tanhd
x x xdx
=
2coth cschd
x xdx
=
Hyperbolic identities
2 2cosh sinh 1x x = 2 2tanh sech 1x x+ = 2 2coth csch 1x x =
sinh( ) sinh cosh cosh sinhx y x y x y =
sinh( ) cosh cosh sinh sinhx y x y x y =
sinh 2 2sinh coshx x x= 2 2cosh 2 cosh sinhx x x= +
2 1 cosh 2sinh
2
xx
+=
2 1 cosh 2cosh2
xx
+=
Inverse Hyperbolic functions
( ) ( )1 2sinh ln 1 ,x x x x = + +
( )
1 2cosh ln 1 [1, )x x x x
= +
( )11 1
tanh ln 1,12 1
xx x
x
+ =
( ) ( )11 1
coth ln , 1 1,2 1
xx x
x
+ =
2
1 1 1sech ln (0,1]x
x xx
+
=
( ) ( )2
1 1 1csch ln ,0 0,
xx x
x x
= +
Inverse Hyperbolic derivates
1
2
1sinh
1
dx
dx x
=
+
1
2
1cosh
1
dx
dx x
=
1
2
1tanh
1
dx
dx x
=
2
1csch
1
dx
dx x x=
+
1
2
1sech
1
dx
dx x x
=
1
2
1coth
1
dx
dx x
=
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Analytic Geometry Formulas
1. Lines in two dimensions
Line forms
Slope - intercept form:
y mx b= + Two point form:
( )2 11 12 1
y yy y x x
x x
=
Point slope form:
( )1 1y y m x x = Intercept form
( )1 , 0x y
a ba b
+ =
Normal form:cos sinx y p + =
Parametric form:
1
1
cos
sin
x x t
y y t
= +
= +
Point direction form:
1 1x x y y
A B
=
where (A,B) is the direction of the line and 1 1 1( , )P x y lies
on the line.
General form:
0 0 0A x B y C A or B + + =
Distance
The distance from 0Ax By C+ + = to 1 1 1( , )P x y is
1 1
2 2
A x B y Cd
A B
+ +=
+
Concurrent linesThree lines
1 1 1
2 2 2
3 3 3
0
0
0
A x B y C
A x B y C
A x B y C
+ + =
+ + =
+ + =
are concurrent if and only if:
1 1 1
2 2 2
3 3 3
0
A B C
A B C
A B C
=
Line segment
A line segment 1 2P P can be represented in parametric
form by
( )
( )
1 2 1
1 2 1
0 1
x x x x t
y y y y t
t
= +
= +
Two line segments 1 2P P and 3 4P P intersect if any only if
the numbers
2 1 2 1 3 1 3 1
3 1 3 1 3 4 3 4
2 1 2 1 2 1 2 1
3 4 3 4 3 4 3 4
x x y y x x y y
x x y y x x y ys and t
x x y y x x y y
x x y y x x y y
= =
satisfy 0 1 0 1s and t
2. Triangles in two dimensions
Area
The area of the triangle formed by the three lines:
1 1 1
2 2 2
3 3 3
0
00
A x B y C
A x B y CA x B y C
+ + =
+ + =
+ + =
is given by
2
1 1 1
2 2 2
3 3 3
2 21 1 3 3
3 32 2 1 1
2
A B C
A B C
A B CK
A BA B A B
A BA B A B
=
The area of a triangle whose vertices are 1 1 1( , )P x y ,
2 2 2( , )P x y and 3 3 3( , )P x y :
1 1
2 2
3 3
11
12
1
x y
K x y
x y
=
2 1 2 1
3 1 3 1
1.
2
x x y yK
x x y y
=
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Centroid
The centroid of a triangle whose vertices are 1 1 1( , )P x y ,
2 2 2( , )P x y and 3 3 3( , )P x y :
1 2 3 1 2 3( , ) ,
3 3
x x x y y yx y
+ + + + =
Incenter
The incenter of a triangle whose vertices are 1 1 1( , )P x y ,
2 2 2( , )P x y and 3 3 3( , )P x y :
1 2 3 1 2 3( , ) ,ax bx cx ay by cy
x ya b c a b c
+ + + + =
+ + + +
where a is the length of2 3
P P , b is the length of1 3
P P ,
and c is the length of1 2.PP
Circumcenter
The circumcenter of a triangle whose vertices are
1 1 1( , )P x y , 2 2 2( , )P x y and 3 3 3( , )P x y :
2 2 2 2
1 1 1 1 1 1
2 2 2 2
2 2 2 2 2 2
2 2 2 2
3 3 3 3 3 3
1 1 1 1
2 2 2 2
3 3 3 3
1 1
1 1
1 1( , ) ,1 1
2 1 2 1
1 1
x y y x x y
x y y x x y
x y y x x yx yx y x y
x y x y
x y x y
+ +
+ +
+ + =
Orthocenter
The orthocenter of a triangle whose vertices are
1 1 1( , )P x y , 2 2 2( , )P x y and 3 3 3( , )P x y :
2 2
1 2 3 1 1 2 3 1
2 2
2 3 1 2 2 3 1 2
2 2
3 1 2 3 3 1 2 3
1 1 1 1
2 2 2 2
3 3 3 3
1 1
1 1
1 1( , ) ,
1 1
2 1 2 1
1 1
y x x y x y y x
y x x y x y y x
y x x y x y y xx y
x y x y
x y x y
x y x y
+ +
+ +
+ + =
3. Circle
Equation of a circle
In an x-y coordinate system, the circle with centre (a, b)
and radius r is the set of all points (x, y) such that:
( ) ( )2 2 2
x a y b r + =
Circle is centred at the origin
2 2 2x y r+ =
Parametric equations
cos
sin
x a r t
y b r t
= +
= +
where t is a parametric variable.
In polar coordinates the equation of a circle is:
( )2 2 2
2 coso or rr r a + =
Area
2A r =
Circumference
2c d r = =
Theoremes:
(Chord theorem)
The chord theorem states that if two chords, CD and EF,intersect at G, then:
CD DG EG FG = (Tangent-secant theorem)
If a tangent from an external point D meets the circle atC and a secant from the external point D meets the circleat G and E respectively, then
2DC DG DE= (Secant - secant theorem)
If two secants, DG and DE, also cut the circle at H and Frespectively, then:
DH DG DF DE = (Tangent chord property)
The angle between a tangent and chord is equal to thesubtended angle on the opposite side of the chord.
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4. Conic Sections
The Parabola
The set of all points in the plane whose distances from afixed point, called the focus, and a fixed line, called thedirectrix, are always equal.
The standard formula of a parabola:2
2y px=
Parametric equations of the parabola:
22
2
x pt
y pt
=
=
Tangent line
Tangent line in a point 0 0( , )D x y of a parabola2
2y px=
( )0 0y y p x x= + Tangent line with a given slope (m)
2
py mx
m= +
Tangent lines from a given point
Take a fixed point 0 0( , )P x y .The equations of the
tangent lines are
( )
( )
0 1 0
0 2 0
2
0 0 01
0
2
0 0 0
1
0
22
2
2
y y m x x and
y y m x x where
y y pxm and x
y y pxm
x
=
=
+ =
=
The Ellipse
The set of all points in the plane, the sum of whosedistances from two fixed points, called the foci, is aconstant.
The standard formula of a ellipse
2 2
2 21
x y
a b+ =
Parametric equations of the ellipse
cos
sin
x a t
y b t
=
=
Tangent line in a point 0 0( , )D x y of a ellipse:
0 0
2 21
x x y y
a b+ =
Eccentricity:
2 2a be
a
=
Foci:
2 2 2 2
1 2
2 2 2 2
1 2
( ,0) ( ,0)
(0, ) (0, )
if a b F a b F a b
if a b F b a F b a
> =>
< =>
Area:
K a b=
The Hyperbola
The set of all points in the plane, the difference of whosedistances from two fixed points, called the foci, remainsconstant.
The standard formula of a hyperbola:2 2
2 21
x y
a b =
Parametric equations of the Hyperbola
sin
sin
cos
ax
t
b ty
t
=
=
Tangent line in a point 0 0( , )D x y of a hyperbola:
0 0
2 21
x x y y
a b =
Foci:
2 2 2 2
1 2
2 2 2 2
1 2
( ,0) ( ,0)
(0, ) (0, )
if a b F a b F a b
if a b F b a F b a
> => + +
< => + +
Asymptotes:
b bif a b y x and y x
a aa a
if a b y x and y xb b
> => = =
< => = =
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5. Planes in three dimensions
Plane forms
Point direction form:
1 1 1x x y y z z
a b c
= =
where P1(x1,y1,z1) lies in the plane, and the direction(a,b,c) is normal to the plane.
General form:
0Ax By Cz D+ + + =
where direction (A,B,C) is normal to the plane.
Intercept form:
1x y z
a b c+ + =
this plane passes through the points (a,0,0), (0,b,0), and(0,0,c).
Three point form
3 3 3
1 3 1 3 1 3
2 3 2 3 2 3
0
x x y y z z
x x y y z z
x x y y z z
=
Normal form:
cos cos cosx y z p + + =
Parametric form:
1 1 2
1 1 2
1 1 2
x x a s a t
y y b s b t
z z c s c t
= + +
= + +
= + +
where the directions (a1,b1,c1) and (a2,b2,c2) areparallel to the plane.
Angle between two planes:
The angle between two planes:
1 1 1 1
2 2 2 2
0
0
A x B y C z D
A x B y C z D
+ + + =
+ + + =
is
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
arccosA A B B C C
A B C A B C
+ +
+ + + +
The planes are parallel if and only if
1 1 1
2 2 2
A B C
A B C= =
The planes are perpendicular if and only if
1 2 1 2 1 20A A B B C C+ + =
Equation of a plane
The equation of a plane through P1(x1,y1,z1) and parallelto directions (a1,b1,c1) and (a2,b2,c2) has equation
1 1 1
1 1 1
2 2 2
0
x x y y z z
a b c
a b c
=
The equation of a plane through P1(x1,y1,z1) andP2(x2,y2,z2), and parallel to direction (a,b,c), has equation
1 1 1
2 1 2 1 2 1 0
x x y y z z
x x y y z z
a b c
=
Distance
The distance of P1(x1,y1,z1) from the plane Ax + By +Cz + D = 0 is
1 1 1
2 2 2
Ax By Cz
d A B C
+ +
=+ +
Intersection
The intersection of two planes
1 1 1 1
2 2 2 2
0,
0,
A x B y C z D
A x B y C z D
+ + + =
+ + + =
is the line
1 1 1 ,x x y y z z
a b c
= =
where
1 1
2 2
B Ca
B C=
1 1
2 2
C Ab
C A=
1 1
2 2
A Bc
A B=
1 1 1 1
2 2 2 2
1 2 2 2
D C D Bb c
D C D B
x a b c
=+ +
1 1 1 1
2 2 2 2
1 2 2 2
D A D Cc c
D A D Cy
a b c
=+ +
1 1 1 1
2 2 2 2
1 2 2 2
D B D Aa b
D B D Az
a b c
=+ +
If a = b = c = 0, then the planes are parallel.
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Limits and Derivatives Formulas
1. Limits
Properties
if lim ( )x a
f x l
= and lim ( )x a
g x m
= , then
[ ]lim ( ) ( )x a
f x g x l m
=
[ ]lim ( ) ( )x a
f x g x l m
=
( )lim
( )x a
f x l
g x m= where 0m
lim ( )x a
c f x c l
=
1 1lim
( )x a f x l= where 0l
Formulas
1lim 1
n
xe
n
+ =
( )1
lim 1 nx
n e
+ =
0
sinlim 1x
x
x=
0
tanlim 1x
x
x=
0
cos 1lim 0x
x
x =
1limn n
n
x a
x ana
x a
=
0
1lim ln
n
x
aa
x
=
2. Common Derivatives
Basic Properties and Formulas
( ) ( )cf cf x =
( ) ( ) ( )f g f x g x = +
Product rule
( )f g f g f g = +
Quotient rule
2
f f g f g
g g
=
Power rule
( ) 1n nd
x nxdx
=
Chain rule
( )( )( ) ( )( ) ( )d
f g x f g x g xdx
=
Common Derivatives
( ) 0d
cdx
=
( ) 1d
xdx
=
( )sin cosd
x x
dx
=
( )cos sind
x xdx
=
( ) 22
1tan sec
cos
dx x
dx x= =
( )sec sec tand
x x xdx
=
( )csc csc cotd
x xdx
=
( )2
2
1
cot cscsin
d
x xdx x= =
( )12
1sin
1
dx
dx x
=
( )12
1cos
1
dx
dx x
=
( )1 21
tan1
dx
dx x
=
+
( ) lnx xd
a a adx
=
( )x xd
e edx
=
( )1
ln , 0d
x xdx x
= >
( )1
ln , 0d
x xdx x
=
( )1
log , 0ln
a
dx x
dx x a= >
-
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3. Higher-order Derivatives
Definitions and properties
Second derivative
2
2
d dy d yf
dx dx dx
=
Higher-Order derivative( ) ( )( )1n nf f
=
( )( ) ( ) ( )n n n
f g f g+ = +
( )( ) ( ) ( )n n n
f g f g =
Leibnizs Formulas
( ) 2 .f g f g f g f g = + +
( ) 3 3f g f g f g f g f g = + + +
( )( ) ( ) ( ) ( ) ( ) ( )1 21
...1 2
n n n n nn nf g f g nf g f g fg
= + + + +
Important Formulas
( )( )
( )
!
!
nm m nm
x xm n
=
( )( )
!n
nx n=
( )( ) ( ) ( )
11 1 !
logln
n
n
a n
nx
x a
=
( )( ) ( ) ( )
11 1 !
ln
n
n
n
nx
x
=
( )( )
lnn
x x na a a=
( )( )n
x xe e=
( )( )
lnn
mx n mx na m a a=
( )( )
sin sin2
n nx x
= +
( )( )
cos cos2
n nx x
= +
-
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Integration Formulas
1. Common Integrals
Indefinite Integral
Method of substitution
( ( )) ( ) ( )f g x g x dx f u du = Integration by parts
( ) ( ) ( ) ( ) ( ) ( )f x g x dx f x g x g x f x dx =
Integrals of Rational and Irrational Functions
1
1
nn x
x dx Cn
+
= ++
1lndx x C
x= +
c dx cx C = + 2
2
xxdx C= +
32
3
xx dx C= +
2
1 1dx C
x x= +
2
3
x xxdx C= +
2
1arctan
1dx x C
x= +
+
2
1arcsin
1dx x C
x= +
Integrals of Trigonometric Functions
sin cosx dx x C= +
cos sinx dx x C= +
tan ln secx dx x C= +
sec ln tan secx dx x x C= + +
( )21
sin sin cos2
x dx x x x C= +
( )21
cos sin cos2
x dx x x x C= + +
2tan tanx dx x x C= +
2sec tanx dx x C= +
Integrals of Exponential and Logarithmic Functions
ln lnx dx x x x C= +
( )
1 1
2ln ln1 1
n n
n
x xx x dx x Cn n
+ +
= ++ +
x xe dx e C = +
ln
xx b
b dx C b
= +
sinh coshx dx x C= +
cosh sinhx dx x C= +
-
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2. Integrals of Rational Functions
Integrals involving ax + b
( )( )
( )( )
1
11
nn ax b
ax b dxa
fo nn
r
++
+ =+
1 1lndx ax b
ax b a= +
+
( )( )
( ) ( )( ) ( )
1
2
11
2,
12
n na n x bx ax b dx ax b
a n nfor n n
+
+ + = +
+ +
2ln
x x bdx ax b
ax b a a= +
+
( ) ( )2 2 2
1ln
x bdx ax b
a ax b aax b= + +
++
( )( )
( )( )( )( )12
12
1,
21n n
a n x bx dx
ax b a n nfor n
ax bn
=+
+
( )( )
222
3
12 ln
2
ax bxdx b ax b b ax b
ax b a
+ = + + + +
( )
2 2
2 3
12 ln
x bdx ax b b ax b
ax baax b
= + + ++
( ) ( )
2 2
3 3 2
1 2ln
2
x b bdx ax b
ax baax b ax b
= + + ++ +
( )
( ) ( ) ( )( )
3 2 122
3
21
3 2 11, 2,3
n n n
n
ax b b a b b ax bxdx
n nfo
nar n
ax b
+ + + = + +
( )
1 1ln
ax bdx
x ax b b x
+=
+
( )2 21 1
lna ax b
dxbx xx ax b b
+= +
+
( ) ( )2 2 2 321 1 1 2
ln
ax b
dx a xb a xb ab x bx ax b
+= +
++
Integrals involving ax2
+ bx + c
2 2
1 1 xdx arctg
a ax a=
+
2 2
1ln
1 2
1ln
2
a xfor x a
a a xdx
x ax afor x a
a x a
+
-
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2
2 2
22
2 2 2
2
2 2arctan 4 0
4 4
1 2 2 4ln 4 0
4 2 4
24 0
2
ax bfor ac b
ac b ac b
ax b b acdx for ac b
ax bx c b ac ax b b ac
for ac bax b
+ >
+ =
+ += + + +
-
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4. Integrals of Logarithmic Functions
ln lncxdx x cx x=
ln( ) ln( ) ln( )b
ax b dx x ax b x ax ba
+ = + + +
( ) ( )2 2
ln ln 2 ln 2x dx x x x x x= +
( ) ( ) ( )1
ln ln lnn n n
cx dx x cx n cx dx
=
( )
2
lnln ln ln
ln !
i
n
xdxx x
x i i
=
= + +
( ) ( )( ) ( )( )
1 11
1
1ln 1 ln lnn n n
for ndx x dx
nx n x x
= +
( )( )1
2
ln 1n
11l
1
m m xx xdx xm m
for m+ = + +
( )( )
( ) ( )1
1lnln
1 11ln
nmn nm m
x x nx x dx x x dx
mr
mfo m
+
= + +
( ) ( )( )
1ln ln
11
n nx x
dx for nx n
+
= +
( )( )
2
lnln0
2
nn xx
dx for nx n
=
( ) ( )( )
1 2 1
ln ln 1
1 11
m m m
x xdx
x m x mfor
xm
=
( ) ( )
( )
( )( )
1
1
ln ln n1
l
11
n n n
m m m
x x xndx dx
mx m x xfor m
= +
ln lnln
dxx
x x=
( )( ) ( )
1
1 lnln ln 1
!ln
i ii
ni
n xdxx
i ix x
=
= +
( ) ( )( ) ( )11
ln 1 ln1
n n
dx
x x nf
xor n
=
( ) ( )2 2 2 2 1ln ln 2 2 tanx
x a dx x x a x aa
+ = + +
( ) ( ) ( )( )sin ln sin ln cos ln2
xx dx x x=
( ) ( ) ( )( )cos ln sin ln cos ln2
xx dx x x= +
-
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5. Integrals of Trig. Functions
sin cosxdx x=
cos sinxdx x=
2 1sin sin 2
2 4
xxdx x=
2 1cos sin 2
2 4
xxdx x= +
3 31sin cos cos
3xdx x x=
3 31cos sin sin
3xdx x x=
ln tansin 2
dx xxdx
x=
ln tancos 2 4
dx x
xdxx
= +
2cot
sin
dxxdx x
x=
2tan
cos
dxxdx x
x=
3 2
cos 1ln tan
sin 2sin 2 2
dx x x
x x= +
3 2
sin 1ln tan
2 2 4cos 2cos
dx x x
x x
= + +
1sin cos cos 2
4x xdx x=
2 31sin cos sin
3x xdx x=
2 31sin cos cos
3x xdx x=
2 2 1sin cos sin 4
8 32
xx xdx x=
tan ln cosxdx x=
2
sin 1
coscos
xdx
xx=
2sin
ln tan sincos 2 4
x xdx x
x
= +
2tan tanxdx x x=
cot ln sinxdx x=
2
cos 1
sinsin
xdx
xx=
2cos
ln tan cossin 2
x xdx x
x= +
2
cot cotxdx x x=
ln tansin cos
dxx
x x=
2
1ln tan
sin 2 4sin cos
dx x
xx x
= + +
2
1ln tan
cos 2sin cos
dx x
xx x= +
2 2tan cot
sin cos
dxx x
x x=
( )( )
( )( )
2 2sin sin
sin sin2 2
m n x m n xmx nxdx
n m nm n
m
+ +
+ =
( )
( )
( )
( )2 2
cos cossin cos
2 2
m n x m n xmx nxdx
n m nm n
m
+
+ =
( )
( )
( )
( )2 2
sin sincos cos
2 2
m n x m n xmx nxdx
m n m nm n
+ = +
+
1cos
sin cos1
nn xx xdx
n
+
= +
1sinsin cos1
nn xx xdxn
+
=+
2arcsin arcsin 1xdx x x x= +
2arccos arccos 1xdx x x x=
( )21
arctan arctan ln 12
xdx x x x= +
( )21
arccot arccot ln 12
xdx x x x= + +
-
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Math Formulas: Definite integrals of rationalfunctions
1.
0
dx
x2 + a2=
2a
2.
0
xp1 dx
1 + x=
sin(p), 0 < p < 1
3.
0
xm
xn + an=
am+1n
n sin[(m + 1)/n], 0 < m + 1 < n
4.
a
0
dx
a2 x2=
2
5.
a
0
a2 x2 dx =
a2
4
6.
a
0
xm (an
xn)p dx =
am+1+np [(m + 1)/n] (p + 1)
n [(m + 1)/n + p + 1]
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Math Formulas: Definite integrals of trigfunctions
Note: In the following formulas all letters are positive.
Basic formulas
1.
/20
sin2 x dx =
/20
cos2 x dx =
4
2.
0
sin(px)
xdx =
/2 p > 00 p = 0/2 p < 0
3.
0
sin2px
x2=
p
2
4.
0
1 cos(px)x2
dx = p
2
5.
0
cos(px) cos(qx)x
dx = lnq
p
6.
0
cos(px) cos(qx)x2
dx =(qp)
2
7.
20
dx
a + b sin x=
2a2 b2
8.
20
dx
a + b cos(x)=
2a2 b2
9.0 sin ax
2
dx =0 cos(ax
2
) dx =
1
2
2a
10.
0
sin xx
dx =
0
cos xx
dx =
2
11.
0
sin3 x
x3dx =
3
8
12.
0
sin4 x
x4dx =
3
13.
0
tan x
xdx =
2
14./20
dxa + b cos x
= arccos(b/a)a2 b2
Advanced formulas
15.
0
sin(mx) sin(nx) dx =
0 m, n integers and m = n/2 m, n integers and m = n
16.
0
cos(mx) cos(nx) dx =
0 m, n integers and m = n/2 m, n integers and m = n
17.
0
sin(mx)
cos(nx) dx = 0 m, n integers and m + n odd2m/(m
2
n2
) m, n integers and m + n even
18
/2sin2m x dx =
/2cos2m x dx =
1 3 5 . . . 2m 1
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19.
/20
sin2m+1 x dx =
/20
cos2m+1 x dx =2 4 6 . . . 2m
1 3 5 . . . 2m + 1
20.
0
sin2p1 x cos2q1 x dx =(p) q
2 (p + q)
21.
0
sin(px)
cos(qx)
x dx =
0 p > q > 0
/2 0 < p < q/4 p = q > 0
22.
0
sin(px) sin(qx)x2
dx =
p/2 0 < p q q/2 p q > 0
23.
0
cos(mx)
x2 + a2dx =
2aema
24.
0
x sin(mx)
x2 + a2dx =
2ema
25.
0
sin(mx)
x (x2 + a2)dx =
2a2
1 ema
26.20
dx(a + b sin x)2
=20
dx(a + b cos x)2
= 2 a(a2 b2)3/2
27.
20
dx
1 2a cos x + a2 =2
1 a2 , 0 < a < 1
28.
0
x sin x dx
1 2a cos x + a2 =
a ln(1 + a) |a| < 1 ln(1 + 1a) |a| > 1
29.
0
cos(mx) dx
1 2a cos x + a2 =am
1 a2 , a2 < 1
30.
0
sin(axn) dx =1
na1/n(1/n) sin
2n, n > 1
31.0
cos(axn) dx = 1na1/n
(1/n) cos 2n
, n > 1
32.
0
sin x
xpdx =
2 (p) sin(p/2), 0 < p < 1
33.
0
cos x
xpdx =
2 (p) cos(p/2), 0 < p < 1
34.
0
sin(ax2) cos(2bx) dx =1
2
2a
cos
b2
a sin b
2
a
35.
0
cos(ax2) cos(2bx) dx =1
2
2a
cos
b2
a+ sin
b2
a
36.
0
dx
1 + tanm xdx =
4
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Math Formulas: Definite integrals ofexponential functions
1.
0
eax cos bxdx =a
a2 + b2
2.
0
eax sin bxdx =b
a2 + b2
3.
0
eax sin bx
xdx = arctan
b
a
4.
0
eax ebx
xdx = ln
b
a
5.
0
eax2
dx =1
2
a
6.
0
eax2
cos bxdx =1
2
a
eb2
4a
7.
e(ax2+bx+c)dx =
2eb24ac
4a
8.
0
xn eaxdx =(n + 1)
an+1
9.
0
xm eax2
dx =m+12
2a(m+1)/2
10.
0
e(ax2+b/x2)dx =
1
2
ae2
ab
11.
0
x dx
ex
1 =
2
6
12.
0
xn1
ex 1dx = (n)
1
1n+
1
2n+
1
3n+
13.
0
x dx
ex + 1=
2
12
14.
0
xn1
ex + 1dx = (n)
1
1n
1
2n+
1
3n
15.
0
sinmx
e2x 1dx =
1
4coth
m
2
1
2m
16.0
11 + x e
x dxx =
17.
0
ex2
ex
xdx =
1
2
18.
0
1
ex 1
ex
x
dx =
19.
0
eax ebx
x sec(px)dx =
1
2ln
b2 + p2
a2 + p2
20.
0
eax ebx
x csc(px)dx = arctan
b
p arctan
a
p
21.0
eax(1
cosx)x2
dx = arccot a a2
ln(a2 + 1)
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Math Formulas: Definite integrals oflogarithmic functions
1. 1
0
xm(ln x)ndx =(1)nn!
(m + 1)n+1, m > 1, n = 0, 1, 2, . . .
2.
10
ln x
1 + xdx =
2
12
3.
10
ln x
1 x dx = 2
6
4.
10
ln(1 + x)
xdx =
2
12
5.
10
ln(1 x)x
dx = 2
6
6. 10
ln x ln(1 + x) dx = 2
2 l n 2
2
12
7.
10
ln x ln(1 x) dx = 2 2
6
8.
0
xp1 ln x
1 + xdx = 2 csc(p) cot(p), 0 < p < 1
9.
10
xm xnln x
dx = lnm + 1
n + 1
10.
0
ex ln x dx =
11.0
e
x
2
ln x dx =
4 ( + 2 ln2)
12.
0
ln
ex + 1
ex 1
dx =
2
4
13.
/20
ln(sin x)dx =
/20
ln(cos x)dx = 2
ln 2
14.
/20
(ln(sin x))2dx =
/20
(ln(cos x))2dx =
2(ln 2)2 +
3
24
15.
0
x ln(sin x)dx = 2
2ln 2
16.
/2
0
sin x ln(sin x)dx = ln 2 1
17.
20
ln(a + b sin x)dx =
20
ln(a + b cos x)dx = 2 ln
a +
a2 b2
18.
0
ln(a + b cos x)dx = ln
a +
a2 b22
19.
0
ln
a2 2ab cos x + b2
dx =
2 ln a a b > 02 ln b b a > 0
20./40 ln(1 + tan x)dx =
8 ln 2
21
2
sec x ln
1 + b cos x
dx =
1 arccos2 a arccos2 b
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Series Formulas
1. Arithmetic and Geometric Series
Definitions:
First term: a1
Nth term: anNumber of terms in the series: n
Sum of the first n terms: Sn
Difference between successive terms: d
Common ratio: q
Sum to infinity: S
Arithmetic Series Formulas:
( )1 1na a n d = +
1 1
2
i ii
a aa +
+=
1
2
nn
a aS n
+=
( )12 1
2n
a n dS n
+ =
Geometric Series Formulas:
11
nna a q
=
1 1i i ia a a +=
1
1
n
n
a q a
S q
=
( )1 11
n
n
a qS
q
=
1
11 1fo
aS
qr q
<
-
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3. Taylor and Maclaurin Series
Definition:
( )( )
( )
( )
112 ( )( )( )( ) ( ) ( ) . . .
2! 1 !
nn
n
f a x af a x af x f a f a x a R
n
= + + + + +
( ) ( )( )
( ) ( )( ) ( )
( )
1
'!
'1 !
nn
n
nn
n
f x aR Lagrange s form a x
n
f x x aR Cauch s form a x
n
=
=
This result holds if f(x) has continuous derivatives of order n at last. If lim 0nn
R
= , the infinite series obtained is called
Taylor series for f(x) about x = a. If a = 0 the series is often called a Maclaurin series.
Binomial series
( )( ) ( )( )1 2 2 3 3
1 2 2 3 3
1 1 2...
2! 3!
...1 2 3
n n n n n
n n n n
n n n n na x a na x a x a x
n n na a x a x a x
+ = + + + +
= + + + +
Special cases:
( )1 2 3 4
1 1 ... 1 1x x x x xx
+ = + + <
-
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Series for trigonometric functions
3 5 7
sin ...3! 5! 7!
x x xx x= + +
2 4 6
cos 1 ...2! 4! 6!
x x xx = + +
( )( )
2 2 2 13 5 7
2 2 12 17tan ...3 15 315 2 ! 2 2
n n n
nB xx x xx xn
x
= + + + + + < <
( )
2 2 13 5 21 2cot ...
3 45 94 20
5 !
n n
nB xx x xxx
xn
= <