Adicione formule

sin2 x+ cos2 x = 1 tgx · ctgx = 1 sin(x± y) = sinxcosy± cosxsiny

tg(x± y) =tgx± tgy

1∓ tgx · tgycos(x± y) = cosxcosy∓ sinxsiny ctg(x± y) =

ctgx · ctgy∓1ctgy± ctgx

sinx = 2sinx2

cosx2

cosx = cos2 x2− sin2 x

2tgx =

2tg x2

1− tg2 x2

ctgx =ctg2 x

2 −12ctg x

2sin2 x =

1− cos(2x)2

cos2 x =1+ cos(2x)

2

tg2 x =1− cos(2x)1+ cos(2x)

ctg2 x =1+ cos(2x)1− cos(2x)

sinx+ siny = 2sinx+ y

2cos

x− y2

sinx− siny = 2cosx+ y

2sin

x− y2

cosx+ cosy = 2cosx+ y

2cos

x− y2

cosx− cosy =−2sinx+ y

2sin

x− y2

Nizovi i limesi nizova

Aritmeticki i geometrijski niz

Za aritmeticki niz an = a1 +(n−1)d, n ∈ N važin

∑k=1

ak =n2(a1 +an) = na1 +

n(n−1)2

d.

Za geometrijski niz bn = b1qn−1, n ∈ N važin

∑k=1

bn =

b11−qn

1−q , q = 1

nb1 , q = 1

∞

∑k=1

bn =

b11

1−q , |q| ≤ 1

divergira , |q| ≥ 1

Tablica limesa nizova

limn→∞

qn =

0 , |q|< 1

1 , q = 1

∞ , q > 1

divergira , q ≤−1

. limn→∞

1nα =

0 , α > 0

1 , α = 1

∞ , α < 0

. limn→∞

nα =

0 , α < 0

1 , α = 1

∞ , α > 0

.

limn→∞

nbqn = 0; |q|< 1, b ∈ R. limn→∞

n√

a = 1; a > 0. limn→∞

n√

n = 1.

limn→∞

nα

n!= 0; α ∈ R. lim

n→∞

an

n!= 0; a ∈ R. lim

n→∞

(1+

1n

)n= e.

limn→∞

an =±∞ ⇒ limn→∞

(1+

1an

)an

= e.

Tablica limesa funkcija

limx→±∞

pnxn + pn−1xn−1 + . . .+ p1x+ p0

qmxm +qm−1xm−1 + . . .+q1x+q0=

0 , n < m

pnqn

, n = m

±∞ , n > m

. limx→±∞

(1+

1x

)x= e. lim

x→0(1+ x)

1x = e.

limx→∞

xα =

0 , α < 0

1 , α = 0

∞ , α > 0

limx→∞

1xα =

0 , α > 0

1 , α = 0

∞ , α < 0

. limx→0

sinxx

= 1.

limx→0

ex −1x

= 1. limx→0

ln(1+ x)x

= 1. limx→0

(1+ x)α −1x

= α; α ∈ R.

Tablica izvoda funkcija

f (x) = α 7→ f ′ (x) = 0, α ∈ R. f (x) = xα 7→ f ′ (x) = αxα−1, α ∈ R. f (x) = ex 7→ f ′ (x) = ex.

f (x) = ax 7→ f ′ (x) = lna ·ax, a > 0. f (x) = lnx 7→ f ′ (x) =1x

. f (x) = loga x 7→ f ′ (x) =1

lna · x, a > 0.

f (x) = sinx 7→ f ′ (x) = cosx. f (x) = cosx 7→ f ′ (x) =−sinx. f (x) = tgx 7→ f ′ (x) =1

cos2 x.

f (x) = cosx 7→ f ′ (x) =−sinx. f (x) = arcsinx 7→ f ′ (x) =1√

1− x2. f (x) = arccosx 7→ f ′ (x) =− 1√

1− x2.

f (x) = arctgx 7→ f ′ (x) =1

1+ x2 . f (x) = arcctgx 7→ f ′ (x) =− 11+ x2 .

Tablica razvoja funkcija u Maklorenov red

ex =∞

∑n=0

xn

n!, x ∈ R. sinx =

∞

∑n=0

(−1)n x2n+1

(2n+1)!, x ∈ R. cosx =

∞

∑n=0

(−1)n x2n

(2n)!, x ∈ R.

ln(1+ x) =∞

∑n=1

(−1)n−1 xn

n, x ∈ (−1,1]. ln(1− x) =−

∞

∑n=1

xn

n, x ∈ [−1,1). (1+ x)m =

m

∑n=0

(mn

)xn, x ∈ R, m ∈ N.

(1+ x)m =∞

∑n=1

(mn

)xn, x ∈ (−1,1), m ∈ N.

11− x

=∞

∑n=0

xn, x ∈ (−1,1).

gde je za m ∈ N i n ∈ N,(m

0

)= 1,

(mn

)=

m(m−1)(m−2) . . .(m−n+1)n!

Tablica integrala

0 ·dx = c.∫

dx = x+ c.

∫xαdx =

xα+1

α+1+ c, α ∈ R\{−1}.

∫ 1x

dx = ln |x|+ c.

∫exdx = ex + c.

∫axdx =

ax

lna+ c, a > 0, a = 1.

∫sinxdx =−cosx+ c.

∫cosxdx = sinx+ c.

∫tgxdx =− ln |cosx|+ c.

∫ctgxdx = ln |sinx|+ c.

∫ 1cos2 x

dx = tgx+ c.∫ 1

sin2 xdx =−ctgx+ c.

∫shxdx = chx+ c.

∫chxdx = shx+ c.

∫thxdx = ln |chx|+ c.

∫cthxdx = ln |shx|+ c.

∫ 1ch2 x

dx = thx+ c.∫ 1

sh2 xdx =−cthx+ c.

∫ 1a2 + x2 dx =

1a

arctgxa+ c, a = 0.

∫ 1a2 − x2 dx =

12a

ln∣∣∣∣a+ xa− x

∣∣∣∣+ c =1a

arcthxa+ c, a = 0.

∫ 1√x2 +a2

dx = ln∣∣∣x+√

x2 +a2∣∣∣+ c = arsh

xa

, a = 0.∫ 1√

x2 −a2dx = ln

∣∣∣x+√x2 −a2

∣∣∣+ c = archxa

, a = 0.

∫ 1√a2 − x2

dx = arcsinxa+ c, a = 0.