ALL MAths Formulas From Maths.org

7/27/2019 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= >


12/25

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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

= +


13/25

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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= +


14/25

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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

+


15/25

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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

+ >

+ =

+ += + + +


16/25

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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= +


17/25

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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= + +


18/25

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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]
http://www.mathportal.org/mathformulas.phphttp://www.mathportal.org/mathformulas.php


19/25

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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


20/25

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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


21/25

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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)


22/25

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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


23/25

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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

<


24/25

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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

+ = + + <


25/25

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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

= <

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