Taula Integrals PDF

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Taula d’integrals Quan la funci´o que volem integrar ´es la derivada d’una de les funcions b`asiques, diem que la integral ´es una integral immediata. Les integrals quasi-immediates es calculen a partir de les integrals immediates i la regla de la cadena Z g ′ (f (x)) f ′ (x) dx = g (f (x)) + c,

c∈R

Integrals immediates

Integrals quasi-immediates

Z

dx = x + c

Z

f ′ (x) dx = f (x) + c

Z

xn dx =

Z

f ′ (x)(f (x))n dx =

Z

1 dx = ln |x| + c x

Z

f ′ (x) dx = ln |f (x)| + c f (x)

Z

ex dx = ex + c

Z

f ′ (x)ef (x) dx = ef (x) + c f ′ (x)af (x) dx =

xn+1 + c, n 6= 1 n+1

(f (x))n+1 + c, n 6= 1 n+1

af (x) +c ln a

Z

a dx = e + c

Z

Z

cos x dx = sin x + c

Z

f ′ (x) cos(f (x)) dx = sin(f (x)) + c

Z

sin x dx = − cos x + c

Z

f ′ (x) sin(f (x)) dx = − cos(f (x)) + c

Z

1 dx = tan x + c cos2 x

Z

f ′ (x) dx = tan(f (x)) + c cos2 (f (x))

Z

f ′ (x)(1 + tan2 f (x)) dx = tan(f (x)) + c

Z

f ′ (x) dx = − cot(f (x)) + c sin2 (f (x))

Z

f ′ (x)(1 + cot2 f (x)) dx = − cot(f (x)) + c

Z Z Z

x

x

(1 + tan2 x) dx = tan x + c 1 2

sin x

dx = − cot x + c 2

(1 + cot x) dx = − cot x + c

Z

1 x √ dx = arcsin + c = 2 2 a a −x

Z

Z

1 x √ dx = − arccos + c a a2 − x2

Z

Z

dx 1 x = arctan + c a2 + x2 a a

Z

Z

dx 1 x−a = arctan +c (x − a)2 + b2 b b

f ′ (x) p

a2

−

(f (x))2

dx = arcsin

f (x) +c = a

p

f (x) f ′ (x) dx = − arccos +c a a2 − (f (x))2

a2

f (x) 1 f ′ (x) dx = arctan +c a + (f (x))2 a...