Calculo 2

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GUIDG.COM – PG. 1 27/9/2010 – CDI-1: TABELA DE INTEGRAIS IMEDIATAS (u, v, w) são variáveis ; (a, b, c) constantes ; e ≈ 2,7182 , ln u = loge u ; sin = seno ; cos = cosseno ; tan = tangente ;
cot = cotangente ; sec = secante ; csc = cossecante ; arc = arco ; h = hiperbólico ; arg = argumento ; s = substituições
IN Integral Imediata IN Integral Imediata

(1) Z du F dv =Z du F Z dv

uffff fffffff fff (5) Z u a du = +c , a ≠@1 a +1
9

`

a

(2) Z b dw = b Z dw

(3) Z du = u + c (7) Z e u du = e u + c
15

a+1

aff fff fff ff (6) Z a u du = +c
u

ln a

(8) Z udv = vu @Z vdu

L M duf ff ff ff f (4) Z =Z u@ 1 du = lnLuM+ c

u

Z sin u du = @ cos u + c Z cos u du = sin u + c Z sec 2 u du = tan u + c Z csc 2 u du = @ cot u + c Z sec u A tan u du = sec u + c Z csc u Acot u du = @ csc u + c

Z sinh u du = cosh u + c Z cosh u du = sinh u + c Z sech u du = tanh u + c
2

10

16

11

17

12

18

Z csch u du = @ coth u + c
2

13

19

Z sech u A tanh u du = @ sech u + c Z csch u A coth u du = @ csch u + c
X

14

20

21

♦Z

fffffff ffffff ffffff fdu fff ff wwww = arc sin u wwww www www www www www ww q1 @ u 2 qa 2 @ u 2

+c

22uf ffdufff fff ffff fff fff fff fff ff f f www w = wwww wwww wwww w www wwww ww w ♦ Z fwwfwwf arc sin f+ c a
Z

\ 1 L1 + u M du ff \ f f ffff f ffff fffM fff f fffff arg tanh u + c , se | u | < 1 ou ffff ffff ff L M+ c = lnL 26 Z =Z Z 2 L1 @ u M arg coth u + c , se | u | > 1 1 @ u2
27

X

L

M

23

1 + u2

du ff ffff ffff ffff ff

= arc tan u + c
f g d e

24

uf fdufff1f fffff ff ff ff f f f f Z fffff= f arc tan f + c 2 2 u +a a a
L M

L fffffff ffffff ffffff fdu fff ff q 2 L www www www www www www www 28 Z wwww = arg cosh u + c = lnLu + u @ 1 2 qu

L M ff ff ff ff ff f fdufff ffff f Z fwfwff arg sinh u + c = lnLu + q u 2 + 1 M + c wwww = ww w ww w ww w ww w ww w ww w L M q1 + u 2 L

wwwwM www www www www www w ww ww

@1

L

29

fff ff ffffduffff fffff ff ffff f Z ffwfwfff @ arg sech | u | + c ww ww = ww ww ww w www www www w w w

wwww M wwww M www www www www www ww M+ c M

u q1 @ u 2

ffffff ff L uffff fdu ff a 1f L ffff fffff fffff ff ff ff f+ ff fffM fa M f = lnL 25 Z 2 M+ c 2

a @u

2a

u @a
M

30

du ffffffff fffffff fffffff ffff Z ffwwwff @ arg csch | u | + c wwww = wwww w ww www www ww ww u q1 + u 2

31

Ztan u du = lnLsec uM+ c
L L M

35



32

Z cot u du = lnLsen uM+ c Z sec u du Z csc u du
L = lnLsec u M + tan uM+ c

36

fffffff f ffdu fff fffffff ffffff ff Z ffwwwwf arc sec u + c wwww = wwww w ww www ww w ww ww

L uf M fffffff ff 1f fffffff f f fffffff f fdu fff ff f f f M Z ffwwwwff farc sinL f + c wwwww = wwwww wwwww wwww wwww wwww L M

u qu 2 @ a 2

a

L M

a

3337

34

L M = lnLcsc u @ cot u M + c

M du ffffffff L ffffffff L fffffff fffffff 2M q 2 wwww wwww wwww wwww wwww wwww www 38 Z wwwww = lnLu + u F a M+ c 2 2 qu

L wwwww M wwwww M wwwww wwww wwww wwww w ww w w L La + q a 2 F u 2 M du 1f L fffffffffff fffffffff fffffffff fffffffff fffffffff f L ffffffffffM f f ffffffffff ffffffffff f M+ c Z wwwww = @ lnL wwwww wwwww wwww wwww wwww w ww w w Ma u qa 2 F u 2 u

u qu 2 @ 1

Fa

L

wwwww wwww M wwww wwww wwww wwww wwww www

S

Substituições e Identidades trigonométricas
I
X wwww wwww wwww wwww wwww wwww www ^ wwww ^q 2 \ a @ u2 ` a ^ ^u = a A sin z Z

(1) sin x =
2

S

II

X wwww wwww wwww wwww www www www ^ wwww ^q 2 \ a + u2 ` a ^ ^u = a A tan z Z

III

X wwww wwww wwww wwww wwww wwww www ^ wwww ^q 2 \ u @ a2` a ^ ^u = a A sec z Z

1fffff2xf @fff ff ffffffff ffcosff f fffff 2

(2) cos 2 x =

1ffcosfff + ff ff fffff2xf ffffff f ffffff 2

(3) sin² x + cos²x = 1 (5) csc² x = 1 + cot² x

(4) sec² x =1 + tan² x

GUIDG.COM – PG. 2
Leg. Notação Descrição e demonstração se possível.

No processo de integração podemos chegar a alguma da formas abaixo, então aplicamos as fórmulas para...
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