Sciaga - Calki

Całki nieoznaczone ′ ∫ f ( x)dx = F ( x) + C ⇒ (F ( x) + C )′ = [ ∫ f (x)dx] = f (x) ∫ a ⋅ f (x)dx = a ∫ f ( x)dx [∫ f (x) ± g( x)]dx = ∫ f ( x)dx ±∫ g (x)dx ∫ f ( x) ⋅ g ′(x)dx = f ( x) ⋅ g (x)∫ f ′(x) ⋅ g ( x)dx ∫ f [ g ( x)] ⋅ g ′(x)dx = ∫ f (t )dt; t = g ( x) ∧ dt = g ′( x)dx Wskazówki przy całk. przez części: 1.∫

xn sin ax n n ax u = x ∫ x e  n

a

a

2.∫ x ln x − u = ln x 3.∫ eax sin bx − u − dow. 4.∫ xn arcsina xdx − u = arcsina x Całki f. wymiernych ln ax + b dx = +C ax + b a (ax + b)n+1 2 +C ∫ (ax + b) dx = a(n + 1) cx + d c ad − bc dx = x + ln ax + b + C ∫ 2 ax + b a a f ′( x)dx = ln f ( x) + C ∫ f ( x) dx 1 x = arctan +C ∫ 2 b b x +b dx 1 x−k = arctan +C ∫ b b ( x − k)2 + b ∫

Całki oznaczone

Całki f. elementarnych ∫ dx = x + C ∫ adx = ax + C x n +1 n +C ∫ x dx = n +1 dx = ln x + C ∫ x ax x +C ∫ a dx = ln a x x ∫ e dx = e + C ∫ sin xdx = − cos x + C ∫ cos xdx = sin x + C ∫ tan xdx = − ln cos x + C ∫ cot xdx = ln sin x + C 1 dx = tan x + C ∫ cos 2 x 1 dx = − cot x + C ∫ 2 sin x 1 1 x dx = arctan + C ∫ 2 a a x + a2 1 x dx = arcsin + C ∫ a a2 − x2

1 dx = ln x + x 2 − a 2 + C x 2 − a2 1 1 a+x dx = ln +C ∫ 2 2a a−x a − x2 Wzory na ... ∫

b

b

P = ∫ f ( x)dx

a a

L = ∫ 1 + [ f ′( x)]2 dx

b ∫ f ( x)dx = [ F ( x)]a = F (b) − F (a)

a b a

∫ f ( x)dx = 0

a b

c

b

a b

a b

c

a

a

∫ f ( x)dx = ∫ f (x)dx + ∫ f ( x)dx; a ≤ c ≤ b ∀

∫ f ( x)dx ≤ ∫ g ( x)dx ⇔ x ∈ a; b f (x) ≤ g ( x)

b

V = π ∫ [ f ( x)]2dx a

b

Pb = 2π ∫ f (x) 1 + [ f ′(x)]2 dx a

Pochodne f. elementarnych c′ = 0 (ax + b)′ = a (ax 2 + bx + c)′ = 2ax + b ( x a )′ = axa −1 1 ( x )′ = 2 x ′  a  = −a   x x2 1 n ′ x = n n x n −1 (sin x)′ = cos x (cos x)′ = − sin x 1 (tan x)′ = cos2 x 1 (cot x)′ = − sin 2 x x′ x a = a ln a ′ ex = ex eax ax ′ e = −? a ′ ( ln x) = 1 x ( loga x)′ = 1 x ln a ( arcsin x)′ = 1 1 − x2 ( arccos x)′ = − 1 1 + x2 ( arctan x)′ = 1 2 1+ x ( arc cot x)′ = − 1 2 1+ x

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