Integral Calculus Formula Sheet 0 PDF

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All the necessary integral formula for calculus ...

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IntegralCalculusFormulaSheet DerivativeRules: d c   0  dx

d x n   nx n 1  dx

d sin x   cos x dx d sec x   sec x tan x dx d tan x   sec 2 x  dx

d cos x    sin x dx d csc x    csc x cot x dx d cot x    csc2 x  dx

d d  cf  x    c dx  f  x  dx

d d d f  x   g  x    dx f  x    dx  g  x  dx

f



d x  a   ax ln a dx d x  e   ex  dx

  f  f g fg     g2 g 

 g   f   g  f  g 

 



d f g  x   f  g  x  g  x  dx





 PropertiesofIntegrals:

 kf (u) du  k f (u) du

  f (u )  g (u ) du  

a

b





f ( x) dx  0 

a

a

c

b

a

f ( x )dx   f ( x )dx  b

c

b

 f ( x)dx   f ( x)dx   f ( x )dx 

f ave 

a a

a



a

b a



f (x )dx  2 f (x )dx iff(x)iseven

a

f (u ) du   g (u) du



1 f ( x )dx  b  a a

f ( x) dx  0iff(x)isodd

a

0

b

f (b)

 g ( f (x )) f (x )dx  a



g (u )du 

f (a )

 udv  uv   vdu 

 IntegrationRules:

 du  u  C n1

u  u du  n 1  C du  u  ln u  C u u  e du  e  C n

1

 a du  ln a a u

 

u

C 

 sin u du   cos u  C  cos u du  sin u  C  sec u du  tan u  C  csc u   cot u  C  csc u cot u du   csc u  C  sec u tan u du  sec u  C  2 2

du 1 u  arctan    C 2 a u a  du u  a 2  u 2  arcsin  a   C u  du 1  u u 2  a 2  a arc sec  a   C

a

2

FundamentalTheoremofCalculus:

F ' x  

d x f t dt  f x  where f t  isacontinuousfunctionon[a,x]. dx  a

b

 f  x  dx  F b   F  a  ,whereF(x)isanyantiderivativeoff(x). a

 RiemannSums: n

b

n

 ca

 c  ai

i

i 1



i 1

n

n

a  b   a  b  i

i

i1

i1

i

i1

n

1  n n

i 1

n

i

2

i 1 n

i i 1

3

ba

 n   height of i th rectangle   width of i th rectangle Right Endpoint Rule:

n( n  1) 2



x 

i 1

i

i 1

i 

n 

a

n

i

n

f ( x) dx  lim  f ( a  ix)x

n

n

 f (a i x )( x )   (

n (n  1)(2n  1) 6

 n (n  1)    2 

2

i 1

i 1

(b  a ) n

) f (a  i

( b  a) n

)

Left Endpoint Rule: n

 i 1

n

f (a  (i  1)x )( x )   ( (b na ) ) f (a  (i 1) (b na ) ) i 1

Midpoint Rule: n



f (a 

i 1



( i 1) i 2

n

  x)( x)   (

( b  a) n

) f (a 

i 1





( i 1)  i ( b a) n 2

)

 NetChange: b



Displacement: v ( x )dx 



DistanceTraveled: v (x ) dx 

a

t

t

b

s( t)  s(0)   v( x) dx  Q (t )  Q(0)   Q( x) dx

a

0

0

 TrigFormulas:

1 cos( x )  cos( x)  sin2 ( x)  cos2 ( x)  1 sin x  sec x   cos x cos x sin(  x)   sin( x)  tan 2 ( x ) 1  sec 2 ( x) x cos2 (x )  12 1  cos(2x )  cot x  cos  csc x  1  sin x sin x

sin 2 (x ) 

1 2

 1 cos(2x ) 

tan x 

 GeometryFomulas: AreaofaSquare: A  s2  

AreaofaTriangle: A  12 bh  

Areaofan EquilateralTrangle: A  43 s 2 

AreaofaCircle: A  r 2 

Areaofa Rectangle: A  bh 

AreasandVolumes: Areaintermsofx(verticalrectangles):

Areaintermsofy(horizontalrectangles):

b

d

 (top  bottom )dx 

 ( right  left) dy 

a

c

GeneralVolumesbySlicing: Given:BaseandshapeofCross‐sections b

DiskMethod: Forvolumesofrevolutionlayingontheaxiswith slicesperpendiculartotheaxis

a d

V     R( x) dx ifslicesarevertical

V   A (x )dx ifslicesarevertical

b

2

a d

V   A ( y )dy ifslicesarehorizontal

V     R( y ) dy ifslicesarehorizontal 2

c

 WasherMethod: Forvolumesofrevolutionnotlayingontheaxiswith slicesperpendiculartotheaxis

c

ShellMethod: Forvolumesofrevolutionwithslicesparalleltothe axis b

b

V    R (x )   r (x )  dx ifslicesarevertical 2

V   2 rhdx ifslicesarevertical

2

a d

V    R ( y )   r ( y ) dy ifslicesarehorizontal 2

2

c

a d

V   2 rhdy ifslicesarehorizontal c

 PhysicalApplications: PhysicsFormulas Mass: Mass=Density*Volume(for3‐Dobjects) Mass=Density*Area(for2‐Dobjects) Mass=Density*Length(for1‐Dobjects)

AssociatedCalculusProblems Massofaone‐dimensionalobjectwithvariablelinear density: b

b

Mass   ( linear density) dx     ( x) dx  distance

a

Work: Work=Force*Distance Work=Mass*Gravity*Distance Work=Volume*Density*Gravity*Distance

a

 Worktostretchorcompressaspring(forcevaries): b

Work 

b

 ( force )dx   F (x )dx  a

a

b



kx 

dx 

a Hooke' s Law for springs

Worktoliftliquid: d

(area of Work   (gravity )(density )(distance )  a slice )dy   c

volume

d

W   9.8*  * A( y )* (a  y )dy (in metric ) c

Force/Pressure: Force=Pressure*Area Pressure=Density*Gravity*Depth

Forceofwaterpressureonaverticalsurface: d

)dy Force   ( gravity )(density )(depth ) ( width   c

area

d

F   9.8*  *(a  y )* w ( y )dy (in metric ) c





IntegrationbyParts:  Knowingwhichfunctiontocalluandwhichtocalldvtakessomepractice.Hereisageneralguide: 1

x ,arccos x, etc )





u



InverseTrigFunction

( sin









LogarithmicFunctions

( log 3x, ln( x 1), etc )









AlgebraicFunctions

( x , x  5,1/ x, etc )









TrigFunctions

( sin(5 x ), tan( x), etc )





dv



ExponentialFunctions



3

x

x

3 3 ( e ,5 , etc )

Functionsthatappearatthetopofthelistaremoreliketobeu,functions atthebottomofthelistaremoreliketobedv.

 TrigIntegrals: Integralsinvolvingsin(x)andcos(x): 1.

2.

3.

Integralsinvolvingsec(x)andtan(x):

Ifthepowerofthesineisoddandpositive: Goal: u  cos x  i. Savea du  sin( x )dx  ii. Converttheremainingfactorsto cos(x ) (using sin2 x  1  cos2 x .)

1.

Ifthepowerofthecosineisoddandpositive: Goal: u  sin x  i. Savea du  cos( x ) dx  ii. Converttheremainingfactorsto sin( x) (using cos 2 x  1 sin 2 x .)

2.

Ifboth sin( x ) and cos( x) haveevenpowers: Usethehalfangleidentities: 2 i. sin ( x)  12  1  cos(2 x ) 

Ifthepowerof sec( x) isevenandpositive: Goal: u  tan x  2

i. Savea du  sec ( x ) dx  ii. Converttheremainingfactorsto tan(x ) (using sec 2 x  1  tan 2 x .)

Ifthepowerof tan( x) isoddandpositive: Goal: u  sec( x )  i. Savea du  sec( x ) tan( x ) dx  ii. Converttheremainingfactorsto sec(x ) (using sec x 1  tan x .) 2



2

Iftherearenosec(x)factorsandthepowerof tan(x)isevenandpositive,use sec x  1  tan x 2

2

2

2

toconvertone tan x to sec x  1  Rulesfor sec(x)andtan(x)alsoworkforcsc(x)and ii. cos ( x)  2 1  cos(2 x)   cot(x)withappropriatenegativesigns Ifnothingelseworks,converteverythingtosinesandcosines. 2

 TrigSubstitution: Expression

Substitution

Domain

a u 

u  a sin  





a2  u 2 

u  a tan 



u a 

u  a sec  

2

2

2

2

2  2

Simplification

   

a 2  u 2  a cos  



a 2  u 2  a sec  

2  2

0    ,  

 2

u 2  a 2  a tan  

  PartialFractions: Linearfactors:

P(x) A B Y Z   m  m 1  m 2  ...  (x  r1 ) ( x  r1 ) ( x  r1 ) ( x  r1 ) ( x  r1 )

Irreduciblequadraticfactors:

P ( x) Ax  B Cx  D Wx  X Yx  Z     ... 2  (x 2  r1 ) m ( x 2  r1) ( x 2  r1) 2 ( x  r1) m1 ( x 2  r1) m

Ifthefractionhasmultiplefactorsinthedenominator,wejustadd thedecompositions. ...