Tablica izvoda: Funkcija ( ) xf Izvod (x)f Ⲡconst c = 0 x 1 x αx ... - Alas
Tablica izvoda: Funkcija ( ) xf Izvod (x)f Ⲡconst c = 0 x 1 x αx ... - Alas
Tablica izvoda: Funkcija ( ) xf Izvod (x)f Ⲡconst c = 0 x 1 x αx ... - Alas
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<strong>Tablica</strong> <strong>izvoda</strong>:<br />
<strong>Tablica</strong> integrala:<br />
<strong>Funkcija</strong> f ( x)<br />
<strong>Izvod</strong> f (x) ′<br />
c = <strong>const</strong><br />
0<br />
x 1<br />
α<br />
x<br />
x<br />
a<br />
x<br />
e<br />
log a x<br />
ln<br />
x<br />
α −1<br />
αx<br />
a x ln a<br />
x<br />
e<br />
1<br />
x ln a<br />
1<br />
x<br />
sin x<br />
cos x<br />
cos x<br />
− sin x<br />
tgx<br />
ctgx<br />
arcsin x<br />
arccos x<br />
arctgx<br />
arcctgx<br />
1<br />
2<br />
cos x<br />
1<br />
−<br />
2<br />
sin x<br />
1<br />
−<br />
1− x<br />
1<br />
2<br />
1− x<br />
1<br />
2<br />
1+ x<br />
1<br />
−<br />
1+ x<br />
2<br />
2<br />
dx = x + c<br />
£<br />
¤ n+<br />
1<br />
n x<br />
x dx = + c<br />
n + 1<br />
= x +<br />
¥<br />
dx<br />
ln c<br />
x<br />
x<br />
e dx = e + c<br />
¦<br />
x a<br />
a dx = + c<br />
ln a<br />
x<br />
sin xdx = − cos x + c<br />
¨<br />
cos xdx = sin x + c<br />
©<br />
dx<br />
= tgx + c<br />
2<br />
cos x<br />
dx<br />
= −ctgx<br />
+ c<br />
2<br />
sin x<br />
x<br />
dx 1 x 1 x<br />
= arctg + c = − arcctg +<br />
<br />
c<br />
2 2<br />
1 , a ≠ 0<br />
x + a a a a a<br />
dx 1 x − a<br />
= ln + c , a ≠ 0<br />
2 2<br />
x − a 2a<br />
x + a<br />
dx<br />
2<br />
= ln x + x ± a + c , a ≠ 0<br />
2 2<br />
x ± a<br />
dx<br />
x<br />
x<br />
= arcsin + c = − arccos + c<br />
<br />
1 , a > 0<br />
2 2<br />
a − x a<br />
a<br />
<br />
2<br />
dx<br />
sin x<br />
x<br />
= ln tg<br />
2<br />
+ c<br />
Površine ravnih figura:<br />
b<br />
t2<br />
P = f ( ¡ x)<br />
dx , P = y(<br />
t)<br />
⋅ x ¢<br />
′(t)<br />
dt ,<br />
a<br />
t1<br />
t<br />
β<br />
1 2<br />
P = ρ ( ϕ)<br />
dϕ<br />
.<br />
2<br />
α<br />
dx x<br />
=<br />
π<br />
ln tg(<br />
+ )<br />
<br />
cos x 2 4<br />
+ c<br />
2 2 x 2 2 a x<br />
a − x dx = a − x + arcsin + c<br />
2<br />
2 a<br />
2<br />
, a > 0<br />
2 x 2 A<br />
x + A dx = x + A + ln x + x + A + c<br />
2 2<br />
<br />
2<br />
b<br />
t2<br />
2<br />
Dužina luka krive: l = 1 + ( f ′(<br />
x))<br />
dx , l = ( x′<br />
( t))<br />
+ ( y′<br />
( t))<br />
dt , l = ρ ( ϕ)<br />
+ ( ρ′<br />
( ϕ))<br />
dϕ<br />
.<br />
a<br />
t1<br />
b<br />
2<br />
Zapremina obrtnih tela: V = π f <br />
2<br />
( x)<br />
dx , V = π y ( t)<br />
⋅ x <br />
π<br />
t′<br />
(t) dt , V =<br />
3<br />
a<br />
t2<br />
t1<br />
t<br />
2<br />
t<br />
2<br />
β<br />
α<br />
β<br />
2 3<br />
α<br />
2<br />
ρ ( ϕ)<br />
sinϕ<br />
dϕ<br />
.<br />
2<br />
Površina omota a obrtnih tela:<br />
b<br />
t2<br />
β<br />
2<br />
2<br />
2<br />
P = 2π f ( x)<br />
1+<br />
( f ′(<br />
x))<br />
dx ,<br />
<br />
2<br />
2<br />
P = 2π y(<br />
t)<br />
( x′<br />
( t))<br />
+ ( y′<br />
( t))<br />
dt , P 2π<br />
ρ(<br />
ϕ)<br />
ρ ( ϕ)<br />
(<br />
<br />
= + ρ′<br />
( ϕ))<br />
sinϕ<br />
dϕ<br />
.<br />
<br />
a<br />
t1<br />
α
e<br />
x<br />
2<br />
n−1<br />
x x x<br />
= 1+<br />
+ + ... + + R<br />
1! 2! ( n −1) !<br />
( x)<br />
, R<br />
n<br />
Maklorenove formule:<br />
x<br />
n!<br />
n<br />
θ x<br />
( x)<br />
= e ,<br />
n < < 1 , x ∈ R<br />
0 θ .<br />
3 5<br />
2n−1<br />
2n+<br />
1<br />
sin x x x<br />
n−1<br />
x<br />
n x<br />
x = − + − ... + ( −1)<br />
+ R ( x)<br />
, R ( x)<br />
= ( −1)<br />
cos θ x , 0 < x ∈ R<br />
2n+<br />
1! 3! 5!<br />
(2n<br />
− 1)!<br />
1 2n<br />
+ 1<br />
(2n<br />
+ 1)!<br />
< θ 1 , .<br />
2 4<br />
2n−2<br />
x x<br />
n−1<br />
x<br />
cos x = 1 − + + ... + ( −1)<br />
+ R2<br />
2! 4!<br />
(2n<br />
− 2)!<br />
2 3 4<br />
n−1<br />
x x x x<br />
n x<br />
ln(1 + x)<br />
= − + − + ... + ( −1)<br />
+ R ( x)<br />
, R<br />
n<br />
1 2 3 4<br />
( n − 1)<br />
(1 + x)<br />
= (<br />
α<br />
) + (<br />
α<br />
) x + (<br />
α<br />
) x<br />
0 1 2<br />
+ ... + (<br />
α<br />
) x<br />
n − 1<br />
α 2<br />
n−1<br />
+ R ( x)<br />
,<br />
n<br />
( 1)...( 1)<br />
(<br />
α α α − α − k +<br />
) =<br />
k<br />
k !<br />
1<br />
1<br />
n<br />
2n<br />
n x<br />
( x)<br />
, R ( x)<br />
= ( −1)<br />
cos θ x , 0 < θ < 1,<br />
x ∈ R .<br />
2n<br />
(2n)!<br />
R<br />
n<br />
( x)<br />
= ( −1)<br />
, α ∈ R , k ∈ N = N } ;<br />
0<br />
∪ { 0<br />
α<br />
n<br />
n+<br />
1<br />
x<br />
n<br />
n (1 + θ x)<br />
n<br />
α −n<br />
( x)<br />
= ( ) x (1 + θ x)<br />
, 1,<br />
n<br />
n−<br />
n n<br />
k k<br />
( −1)<br />
x<br />
α = 1: = ( −1)<br />
x + R ( x)<br />
R ( x)<br />
=<br />
, 0 < θ < 1,<br />
x < 1.<br />
n n<br />
n+<br />
1<br />
1 + x k = 0<br />
(1 + θ x)<br />
n<br />
, 0 < θ < 1 , −1<br />
< x ≤ 1, n > 1.<br />
0 < θ < x < 1,<br />
Trigonometrija:<br />
sin( x + y)<br />
= sin x cos y + cos x sin y<br />
cos( x + y)<br />
= cos x cos y − sin x sin y<br />
tgx + tgy<br />
tg(<br />
x + y)<br />
=<br />
1−<br />
tgx ⋅tgy<br />
ctgxctgy −1<br />
ctg(<br />
x + y)<br />
=<br />
ctgx + ctgy<br />
x + y x − y<br />
sin x + sin y = 2 sin cos<br />
2 2<br />
x + y x − y<br />
cos x + cos y = 2 cos cos<br />
2 2<br />
sin( x + y)<br />
tgx + tgy =<br />
cos x cos y<br />
sin( x + y)<br />
ctgx + ctgy =<br />
sin x sin y<br />
sin 2x<br />
= 2 sin x cos x<br />
cos 2x<br />
= cos x − sin<br />
2tgx<br />
tg2x<br />
=<br />
2<br />
1−<br />
tg x<br />
ctg x −1<br />
ctg2x<br />
=<br />
2ctgx<br />
1 cos<br />
sin 2 x − x<br />
=<br />
2 2<br />
1 cos<br />
cos 2 x + x<br />
=<br />
2 2<br />
2<br />
2<br />
2<br />
x<br />
x<br />
2tg<br />
sin x =<br />
2<br />
2 x<br />
1+<br />
tg<br />
2<br />
2 x<br />
1−<br />
tg<br />
cos x =<br />
2<br />
2 x<br />
1+<br />
tg<br />
2<br />
sin( x − y)<br />
= sin x cos y − cos x sin y<br />
cos( x − y)<br />
= cos x cos y + sin x sin y<br />
tgx − tgy<br />
tg(<br />
x − y)<br />
=<br />
1+<br />
tgx ⋅ tgy<br />
ctgxctgy + 1<br />
ctg(<br />
x − y)<br />
=<br />
ctgy − ctgx<br />
x − y x + y<br />
sin x − sin y = 2 sin cos<br />
2 2<br />
x + y x − y<br />
cos x − cos y = −2 sin sin<br />
2 2<br />
sin( x − y)<br />
tgx − tgy =<br />
cos x cos y<br />
sin( y − x)<br />
ctgx − ctgy =<br />
sin x sin y<br />
1<br />
sin x cos y = [ sin( x − y)<br />
+ sin( x + y)<br />
]<br />
2<br />
1<br />
sin x sin y = [ cos( x − y)<br />
− cos( x + y)<br />
]<br />
2<br />
1<br />
cos x cos y = cos( x − y)<br />
+ cos( x + y)<br />
2<br />
[ ]<br />
sin<br />
cos<br />
2<br />
2<br />
2<br />
tg x<br />
x =<br />
2<br />
1+<br />
tg x<br />
1<br />
x =<br />
1+<br />
tg<br />
2<br />
x