Tabela de Derivadas e Integrais

[pic 1][pic 2]

(01) y = c ⇒ y’= 0

(02) y = u ⇒ y’ = 1

(03) y = c.u ⇒ y’= c.u’

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1. ∫du = u + C

2. du = ln u + C[pic 3][pic 4][pic 5]

u

n        un+1[pic 6]

(04) y = u + v ⇒ y’= u’ + v’

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1. ∫ u du = n +1 + C

(05) y = u . v ⇒ y’= u’. v + u. v’

(06) y = u / v ⇒ y’= u’v- u v’ / v2

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1. ∫a

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u

du =                + C lna[pic 7][pic 8]

(07) y = un ⇒ y’ = n. un -1.u’

′

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1. ∫eu du = eu + C

2. ∫senu du = − cos u + C

1. y = au ⇒ y’ = au lna.u

1. y = eu ⇒ y’ = eu.u ′

(10) y = logau   ⇒  y' = (u' / u).logae

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1. ∫cosu du = sen u + C

2. ∫ tgu du = ln secu + C[pic 9][pic 10]

3. ∫cotgu du = ln senu + C[pic 11][pic 12]

(11) y = ln u ⇒ y’ = (u ′/ u)

1. y = uv ⇒ y’ = v. uv -1. u ′+ uv . ln u.v’

1. y = sen u ⇒ y’= cos u. u ′

1. y = cos u ⇒ y’ = - sen u. u ′

1. y = tg u  ⇒ y’ = sec2 u. u ′

1. y = cotg u ⇒ y’= - cossec2 u. u ′

1. y = sec u  ⇒ y’ = sec u. tg u. u ′

1. y = cossec u ⇒ y’= - cossec u. cotg u. u ′

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1. ∫cosecu du = ln cossecu −cotgu + C

2. ∫secu du = ln secu + tgu + C[pic 13][pic 14][pic 15][pic 16]

3. ∫sec2u du = tgu + C

4. ∫cosec2u du = - cotgu + C

5. ∫secu.tgu du = secu + C

6.          du        = arc sen u + C[pic 17][pic 18]

a

1. ∫        du        = 1 arc tg u + C

1. y = arc sen u ⇒ y’ = u ′ /[pic 19]

1. y = arc cos u ⇒ y’ = - u ′ /[pic 20]

(21) y = arc tg u ⇒ y’ = u ′ / (1 + u2 )

1. y = arc cotg u ⇒ y’ = - u ′ / (1 + u2 )

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(17) ∫

(18) ∫

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a 2 + u 2

du[pic 21]

du

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a        a

= 1 arc sec        + C a[pic 22][pic 23]

= ln u +[pic 24][pic 25][pic 26]

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• C

1. y = arc sec u ⇒ y’ = u ′ / u .[pic 27][pic 28]

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u 2 −1

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(19) ∫        du        = 1 ln u + a + C

(24) y = arc cossec u ⇒ y’= - u ′ / u .[pic 29][pic 30][pic 31][pic 32][pic 33]

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u 2 −1

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a 2 − u 2

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2a        u − a

(20) ∫        du        = − 1 ln        + C

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