Tablica izvoda:
Tablica integrala:
Funkcija f (x )
Izvod f ′(x)
c = const
0
x
1
dx = x + c x n dx =
dx = ln x + c x
xα
x n +1 +c n +1
αxα −1
e x dx = e x + c a
x
e
x
x
a ln a
ax +c ln a
e
a x dx =
x
log a x
1 x ln a
ln x
1 x
sin x
cos x
sin xdx = − cos x + c
cos xdx = sin x + c
dx
− sin x
cos x
dx
1
tgx
cos x 1 − sin 2 x 1
arcsin x
arctgx
dx
1− x 1
dx
x −a dx 2
1 x 1 x arctg + c = − arcctg + c1 , a ≠ 0 a a a a
2
=
1 x−a ln +c, a ≠ 0 2a x+a
x2 ± a 2
2
dx
1+ x 1 − 1 + x2 2
arcctgx
=
x2 + a2
1 − x2 1
−
arccos x
= −ctgx + c
sin 2 x
2
ctgx
= tgx + c
cos 2 x
a −x 2
2
= ln x + x 2 ± a 2 + c , a ≠ 0
= arcsin
dx x = ln tg +c sin x 2
dx x π = ln tg ( + ) + c cos x 2 4
a 2 − x 2 dx =
x a2 x a2 − x2 + arcsin + c , a > 0 2 2 a
x 2 + A dx =
x 2
Površine ravnih figura: β
t2
b
f ( x) dx , P =
P=
y(t ) ⋅ x t′ (t)dt , P =
α
t1
a
1 ρ 2 (ϕ )dϕ . 2
( x t′ (t )) 2 + ( y t′ (t )) 2 dt , l =
1 + ( f ′( x)) 2 dx , l =
Dužina luka krive: l =
t2
b
Zapremina obrtnih tela: V = π f 2 ( x)dx , V = π y 2 (t ) ⋅ xt′ (t)dt , V =
a
A ln x + x 2 + A + c 2
ρ 2 (ϕ ) + ( ρ ′(ϕ ))2 dϕ .
α
t1
a
x2 + A +
β
t2
b
x x + c = − arccos + c1 , a > 0 a a
t1
2π 3
β
ρ 3 (ϕ ) sin ϕ dϕ .
α
Površina omota a obrtnih tela:
t2
b
a
β
f ( x ) 1 + ( f ′( x)) dx , P = 2π
P = 2π
y (t ) ( x′(t )) 2 + ( y ′(t )) 2 dt , P = 2π ρ (ϕ ) ρ 2 (ϕ ) + ( ρ ′(ϕ )) 2 sin ϕ dϕ .
2
t1
α
Maklorenove formule: e x = 1+
x x2 x n −1 xn θ x + + ... + + R n ( x ) , R n ( x) = e , 0 < θ < 1, x ∈ R . 1! 2 ! (n − 1) ! n!
sin x =
x x3 x5 x 2 n −1 x 2 n +1 − + − ... + (−1) n −1 + R2 n +1 ( x) , R 2 n +1 ( x) = (−1) n cos θ x , 0 < θ < 1, x ∈ R . 1! 3 ! 5! (2 n − 1) ! (2n + 1) !
cos x = 1 −
x2 x4 x 2 n− 2 x 2n + + ... + (−1) n −1 + R2 n ( x) , R 2 n ( x ) = (−1) n cos θ x , 0 < θ < 1, x ∈ R . 2! 4! (2n − 2) ! ( 2n) !
ln(1 + x) =
xn x x2 x3 x4 x n −1 , 0 < θ < 1 , − 1 < x ≤ 1 , n > 1. − + − + ... + ( −1) n + R n ( x) , R n ( x) = (−1) n +1 1 2 3 4 ( n − 1) n (1 + θ x) n
α n α −n (1 + x ) α = ( α ) + ( α ) x + ( α ) x 2 + ... + ( α ) x n −1 + Rn ( x ) , R n ( x) = ( ) x (1 + θ x ) , 0 < θ < 1 , n 0 1 2 n −1 (α ) = k
α (α − 1)...(α − k + 1)
1 = 1+ x
α = 1:
k! n −1 k =0
, α ∈ R , k ∈ N 0 = N ∪ { 0} ;
(−1) k x k + Rn ( x) R n ( x ) =
(−1) n x n , 0 < θ < 1, (1 + θ x) n +1
x < 1.
Trigonometrija: sin( x + y ) = sin x cos y + cos x sin y
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
cos( x + y ) = cos x cos y − sin x sin y tg ( x + y ) =
tgx + tgy 1 − tgx ⋅ tgy
ctg ( x + y ) =
ctgxctgy − 1 ctgx + ctgy
x+ y x− y 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
x− y x+ y 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 x + sin y = 2 sin
ctgx + ctgy =
sin x − sin y = 2 sin
sin( x + y ) sin x sin y
ctgx − ctgy =
sin( y − x) sin x sin y
sin 2 x = 2 sin x cos x
1 [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
cos 2 x = cos 2 x − sin 2 x 2tgx tg 2 x = 1 − tg 2 x ctg 2 x =
sin x cos y =
ctg 2 x − 1 2ctgx
sin 2
x 1 − cos x = 2 2
cos 2
x 1 + cos x = 2 2
sin x =
2tg
x 2
x 1 + tg 2 x 1 − tg 2 2 cos x = x 1 + tg 2 2 2
sin 2 x =
cos 2 x =
tg 2 x 1 + tg 2 x
1 1 + tg 2 x
x < 1,
