Integrali Formule

INTEGRALI Tabli ni integrali:

∫ 0 ⋅ dx = C ∫ dx = x + C ∫x ∫

a

1 x a +1 + C (a + 1)

dx =

dx = ln x + C x

∫e

x

dx = e x + C ax +C ln a

x ∫ a dx =

∫ cos xdx = sin x + C ∫ sin xdx = − cos x + C dx

∫ cos

2

x

dx

∫ sin ∫

2

x dx

= tan x + C = − cot x + C

= arcsin x + C 1− x2 dx ∫ 1 + x 2 = arctan x + C dx 1 1+ x ∫ 1 − x 2 = 2 ln 1 − x + C dx 2 2 ∫ x 2 ± a 2 = ln x + x ± a + C xdx 2 2 ∫ x2 ± a2 = x ± a + C xdx 1 2 2 ∫ x 2 ± a 2 = 2 ln x ± a + C 1 dx x−a ∫ x 2 − a 2 = 2a ln x + a + C

∫ sinh xdx = cosh x + C ∫ cosh x = sinh x + C 1

∫ cosh

2

x

1

∫ sinh

2

x

dx = tanh x + C dx = − coth x + C

Parcijalna integracija: ∫ v ⋅ du u,v su proizvoljne funkcije

∫ u ⋅ dv = u ⋅ v −

cos x   x cos x     x   x I tip:  xe  ⇒ u = x dv = e dx   x sin x   sin x      x arccos x   arccos x    II tip:  x ln x  ⇒ u = ln x  x arcsin x   arcsin x   

  dv = xdx   

III tip: integrali u parovima npr.: I1 = ∫ e x cos x dx u = e x dv = cos x dx I + I 2 = ... ⇒ 1   sin x  − I1 + I 2 = ... I 2 = ∫ e x sin x dx 