# tablice matematika aha evo efektan naslov, Beleške' predlog Material Thermodynamics

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Microsoft Word - Tablica.doc

Tablica izvoda:

Funkcija( )xf Izvod(x)f

constc =

0

x

1

αx 1−ααx

xa aa x ln

xe xe

xalog ax ln

1

x ln x

1

xsin

xcos

xcos

xsin−

tgx x2cos

1

ctgx x2sin

1−

xarcsin 21

1

x

xarccos 21

1

x− −

arctgx 21

1

x+

arcctgx 21

1

x+ −

Površine ravnih figura:

= b

a

dxxfP )( ,  ′⋅= 2

1

)(

t

t

t dt(t)xtyP , = β

α

ϕϕρ dP )( 2

1 2 .

Tablica integrala:



+= cxdx

c n

x dxx

n n +

+ =

 +

1

1



+= c x x

dx ln



+= cedxe xx

c a

a dxa

x x +=



ln



+−= cxxdx cossin

+= cxxdx sincos

ctgx x

dx +=

2cos

cctgx x

dx +−=

2sin

122

11 c

a

x arcctg

a c

a

x arctg

aax

dx +−=+= +

, 0≠a

c ax

ax

aax

dx + + −=

ln 2

1 22 , 0≠a

c axx ax

dx +±+= ±



22

22 ln , 0≠a

1 22

arccosarcsin c a

x c

a

x

xa

dx +−=+= −



, 0>a

c x

tg x

dx += 

2 ln

sin

c x

tg x

dx ++= 

) 42

(ln cos

π

c a

xa xa

x dxxa ++−=−



arcsin 22

2 2222 , 0>a

c Axx A

Ax x

dx Ax +++++=+ 

222 ln 22

Dužina luka krive:dxxfl b

a



′+= 2))((1 , dttytxl t

t

tt



′+′= 2

1

22 ))(())(( , ϕϕρϕρ β

α

dl  ′+= 22 ))(()( .

Zapremina obrtnih tela: = b

a

dxxfV )(2π ,  ′⋅= 2

1

)(2 t

t

t dt(t) xtyV π , = β

α

ϕϕϕρπ d V sin)( 3

2 3 .

Površina omota  a obrtnih tela:



′+= b

a

dxxfxfP 2))((1)(2π , dt tytxtyP t

t

′+′= 2

1

22 ))(())(()(2π , ϕϕϕρϕρϕρπ β

α

d P sin))(()()(2 22

′+= .

Maklorenove formule:

)( !)1(

... !2!1

1 12

xR n

x

x

x e n

n x +

− ++++=

− , x

! )( θe

n

x xR

n

n = , Rx ∈<< ,10 θ .

)( ! )12(

)1(... ! 5! 3! 1

sin 12

12 1

53

xR n

xxxx x

n

n n

+

− − +

− −+−+−= , x

n

x xR

n n

n cos

! )12( )1()(

12

12 θ

+ −=

+

+ , Rx ∈<< ,10 θ .

)( ! )22(

)1(... ! 4! 2

1cos 2

22 1

42

xR n

xxx x

n

n n +

− −+++−=

− − , x

n

x xR

n n

n cos

! )2( )1()(

2

2 θ−= , Rx ∈<< ,10 θ .

)( )1(

)1(... 4321

)1ln( 1432

xR n

xxxxx x

n

n n +

− −++−+−=+

, n

n n

n xn

x xR

) 1( )1()( 1

θ+ −= + , 1110 ≤<−<< x ,θ , .1>n

)() 1 (...) 2 () 1 () 0 ()1( 12 xRxnxxx n

n +−++++=+ −ααααα , nn

n xxnxR

−+= αθα ) 1() ()( , , 10 << θ 1< x ,

!

)1)...(1( ) (

k

k k

+−−= αααα , R∈α , } 0 {0 ∪=∈ NNk ;

:1=α )()1( 1

1 1

0

xRx x n

kk n

k

+−= + �

= 1n x)1(

)1( )(

++ −=

θ

nn

n

x xR , , 10 << θ 1 <x .

Trigonometrija:

yxyxyx sincoscossin)sin( +=+ yxyxyx sinsincoscos)cos( −=+

tgytgx

tgytgx yxtg

⋅− +

=+ 1

)(

ctgyctgx

ctgxctgy yxctg

+ −=+ 1)(

yxyxyx sincoscossin)sin( −=− yxyxyx sinsincoscos)cos( +=−

tgytgx

tgytgx yxtg

⋅+ −=−

1 )(

ctgxctgy

ctgxctgy yxctg

− +=− 1)(

2 cos

2 sin2sinsin

yxyx yx

−+ =+

2 cos

2 cos2coscos

yxyx yx

−+ =+

yx

yx tgytgx

coscos

)sin( +=+

yx

yx ctgyctgx

sinsin

)sin( + =+

2 cos

2 sin2sinsin

yxyx yx

+− =−

2 sin

2 sin2coscos

yxyx yx

−+ −=−

yx

yx tgytgx

coscos

)sin( − =−

yx

xy ctgyctgx

sinsin

)sin( −=−

xxx cossin22sin = xxx 22 sincos2cos −=

xtg

tgx xtg

21

2 2

− =

ctgx

xctg xctg

2

1 2

2 −=

[ ])sin()sin( 2

1 cossin yxyxyx ++−=

[ ])cos()cos( 2

1 sinsin yxyxyx +−−=

[ ])cos()cos( 2

1 coscos yxyxyx ++−=

2

cos1

2 sin 2

xx −=

2

cos1

2 cos 2

xx +=

2 1

2 2

sin 2 xtg

x tg

x +

=

2 1

2 1

cos 2

2

x tg

x tg

x +

− =

xtg

xtg x

2

2 2

1 sin

+ =

xtg x

2 2

1

1 cos

+ =

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