Tablica izvoda: Funkcija ( ) xf Izvod (x)f ′ const c = 0 x 1 x αx ... - Alas

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Tablica izvoda:
Tablica integrala:
Funkcija f ( x)
Izvod f (x) ′
c = const
0
x 1
α
x
x
a
x
e
log a x
ln
x
α −1
αx
a x ln a
x
e
1
x ln a
1
x
sin x
cos x
cos x
− sin x
tgx
ctgx
arcsin x
arccos x
arctgx
arcctgx
1
2
cos x
1
−
2
sin x
1
−
1− x
1
2
1− x
1
2
1+ x
1
−
1+ x
2
2
dx = x + c
£
¤ n+
1
n x
x dx = + c
n + 1
= x +
¥
dx
ln c
x
x
e dx = e + c
¦
x a
a dx = + c
ln a
x
sin xdx = − cos x + c
¨
cos xdx = sin x + c
©
dx
= tgx + c
2
cos x
dx
= −ctgx
+ c
2
sin x
x
dx 1 x 1 x
= arctg + c = − arcctg +
c
2 2
1 , a ≠ 0
x + a a a a a
dx 1 x − a
= ln + c , a ≠ 0
2 2
x − a 2a
x + a
dx
2
= ln x + x ± a + c , a ≠ 0
2 2
x ± a
dx
x
x
= arcsin + c = − arccos + c
1 , a > 0
2 2
a − x a
a
2
dx
sin x
x
= ln tg
2
+ c
Površine ravnih figura:
b
t2
P = f ( ¡ x)
dx , P = y(
t)
⋅ x ¢
′(t)
dt ,
a
t1
t
β
1 2
P = ρ ( ϕ)
dϕ
.
2
α
dx x
=
π
ln tg(
+ )
cos x 2 4
+ c
2 2 x 2 2 a x
a − x dx = a − x + arcsin + c
2
2 a
2
, a > 0
2 x 2 A
x + A dx = x + A + ln x + x + A + c
2 2
2
b
t2
2
Dužina luka krive: l = 1 + ( f ′(
x))
dx , l = ( x′
( t))
+ ( y′
( t))
dt , l = ρ ( ϕ)
+ ( ρ′
( ϕ))
dϕ
.
a
t1
b
2
Zapremina obrtnih tela: V = π f
2
( x)
dx , V = π y ( t)
⋅ x
π
t′
(t) dt , V =
3
a
t2
t1
t
2
t
2
β
α
β
2 3
α
2
ρ ( ϕ)
sinϕ
dϕ
.
2
Površina omota a obrtnih tela:
b
t2
β
2
2
2
P = 2π f ( x)
1+
( f ′(
x))
dx ,
2
2
P = 2π y(
t)
( x′
( t))
+ ( y′
( t))
dt , P 2π
ρ(
ϕ)
ρ ( ϕ)
(
= + ρ′
( ϕ))
sinϕ
dϕ
.
a
t1
α

e
x
2
n−1
x x x
= 1+
+ + ... + + R
1! 2! ( n −1) !
( x)
, R
n
Maklorenove formule:
x
n!
n
θ x
( x)
= e ,
n < < 1 , x ∈ R
0 θ .
3 5
2n−1
2n+
1
sin x x x
n−1
x
n x
x = − + − ... + ( −1)
+ R ( x)
, R ( x)
= ( −1)
cos θ x , 0 < x ∈ R
2n+
1! 3! 5!
(2n
− 1)!
1 2n
+ 1
(2n
+ 1)!
< θ 1 , .
2 4
2n−2
x x
n−1
x
cos x = 1 − + + ... + ( −1)
+ R2
2! 4!
(2n
− 2)!
2 3 4
n−1
x x x x
n x
ln(1 + x)
= − + − + ... + ( −1)
+ R ( x)
, R
n
1 2 3 4
( n − 1)
(1 + x)
= (
α
) + (
α
) x + (
α
) x
0 1 2
+ ... + (
α
) x
n − 1
α 2
n−1
+ R ( x)
,
n
( 1)...( 1)
(
α α α − α − k +
) =
k
k !
1
1
n
2n
n x
( x)
, R ( x)
= ( −1)
cos θ x , 0 < θ < 1,
x ∈ R .
2n
(2n)!
R
n
( x)
= ( −1)
, α ∈ R , k ∈ N = N } ;
0
∪ { 0
α
n
n+
1
x
n
n (1 + θ x)
n
α −n
( x)
= ( ) x (1 + θ x)
, 1,
n
n−
n n
k k
( −1)
x
α = 1: = ( −1)
x + R ( x)
R ( x)
=
, 0 < θ < 1,
x < 1.
n n
n+
1
1 + x k = 0
(1 + θ x)
n
, 0 < θ < 1 , −1
< x ≤ 1, n > 1.
0 < θ < x < 1,
Trigonometrija:
sin( x + y)
= sin x cos y + cos x sin y
cos( x + y)
= cos x cos y − sin x sin y
tgx + tgy
tg(
x + y)
=
1−
tgx ⋅tgy
ctgxctgy −1
ctg(
x + y)
=
ctgx + ctgy
x + y x − y
sin x + sin y = 2 sin cos
2 2
x + y x − y
cos x + cos y = 2 cos cos
2 2
sin( x + y)
tgx + tgy =
cos x cos y
sin( x + y)
ctgx + ctgy =
sin x sin y
sin 2x
= 2 sin x cos x
cos 2x
= cos x − sin
2tgx
tg2x
=
2
1−
tg x
ctg x −1
ctg2x
=
2ctgx
1 cos
sin 2 x − x
=
2 2
1 cos
cos 2 x + x
=
2 2
2
2
2
x
x
2tg
sin x =
2
2 x
1+
tg
2
2 x
1−
tg
cos x =
2
2 x
1+
tg
2
sin( x − y)
= sin x cos y − cos x sin y
cos( x − y)
= cos x cos y + sin x sin y
tgx − tgy
tg(
x − y)
=
1+
tgx ⋅ tgy
ctgxctgy + 1
ctg(
x − y)
=
ctgy − ctgx
x − y x + y
sin x − sin y = 2 sin cos
2 2
x + y x − y
cos x − cos y = −2 sin sin
2 2
sin( x − y)
tgx − tgy =
cos x cos y
sin( y − x)
ctgx − ctgy =
sin x sin y
1
sin x cos y = [ sin( x − y)
+ sin( x + y)
]
2
1
sin x sin y = [ cos( x − y)
− cos( x + y)
]
2
1
cos x cos y = cos( x − y)
+ cos( x + y)
2
[ ]
sin
cos
2
2
2
tg x
x =
2
1+
tg x
1
x =
1+
tg
2
x