Formule Integrali I Derivacije

Vrijednosti sinusa i kosinusa ϕ sin ϕ cos ϕ

0 0 1

π 6 1 √2 3 2

π √4 2 √2 2 2

π √3 3 2 1 2

Tablica derivacija

π 2

f (x)

f 0 (x)

f (x)

1 0

xa

axa−1

ln x

sin x

cos x

loga x

cos x

− sin x

sh x

f 0 (x) 1 x 1 x ln a ch x

ch x

sh x

Adicijski teoremi sin(x ± y) cos(x ± y)

= sin x cos y ± cos x sin y = cos x cos y ∓ sin x sin y

tg(x ± y)

=

tg x±tg y 1∓tg x tg y

ctg(x ± y)

=

ctg x ctg y∓1 ctg y±ctg x

Funkcije viˇ sestrukih argumenata sin 2x cos 2x

= 2 sin x cos x = cos2 x − sin2 x

tg 2x

=

ctg 2x

=

2 tg x 1−tg2 x ctg2 x−1 2 ctg x

1 cos2 x 1 ctg x − 2 sin x 1 √ arcsin x 1 − x2 1 arccos x − √ 1 − x2 1 arctgx 1 + x2 1 arcctgx − 1 + x2 tg x

ex x

sin x cos y = 12 (sin(x + y) + sin(x − y)) cos x cos y = 12 (cos(x + y) + cos(x − y)) sin x sin y = 12 (cos(x − y) − cos(x + y)) x−y sin x + sin y = 2 sin x+y 2 cos 2

sin x − sin y = 2 cos

sin

x−y 2

x−y cos x + cos y = 2 cos x+y 2 cos 2 x−y cos x − cos y = −2 sin x+y 2 sin 2

Funkcije poloviˇ cnih argumenata sin2

x 2 cos2 x2

= =

1−cos x 2 1+cos x 2

Neke vaˇ zne formule sin2 x

=

cos2 x = sin x

=

cos x

=

tg2 x 1+tg2 x 1 1+tg2 x 2 tg x 2 1+tg2 x 2 2 x 1−tg 2 1+tg2 x 2

cthx arshx archx arthx arcthx

x

a

Formule pretvorbe

x+y 2

ex

thx

a ln a

1 ch2 x 1 − 2 sh x 1 √ 1 + x2 1 √ x2 − 1 1 1 − x2 1 1 − x2

Tablica integrala R R R R R R R R R R R R R R R R

dx x

= ln |x| + C

xα dx =

xα+1 α+1 x

a ln a

ax dx =

+ C, α ∈ R \ {−1}

+C

ex dx = ex + C sin xdx = − cos x + C cos xdx = sin x + C dx sin2 x

= − ctg x + C

dx sin2 x

= tg x + C

dx x2 +a2 dx x2 −a2

1 a

arctg( xa ) + C, a > 0 ¯ ¯ ¯ ¯ 1 = 2a ln ¯ x−a x+a ¯ + C, a > 0

=

√ dx a2 −x2 √ dx x2 +A

= arcsin( xa ) + C, a > 0 √ = ln |x + x2 + A| + C, A 6= 0

sh xdx = ch x + C ch xdx = sh x + C dx sh2 x

= − cth x + C

dx ch2 x

= th x + C