Integrais. A integral indefinida de uma função f(t) é representada como. Por outro lado, a integral definida, representada como

Integrais

A integral indefinida de uma função f(t) é representada como

∫

f

(

τ d

)

⋅

τ

Por outro lado, a integral definida, representada como

∫

b

τ

⋅

τ

a

f

(

)

d

,

∫

−∞

τ

⋅

τ

b 3

(

)

d

f

_ou

∫

∞

τ

⋅

τ

a

f

(

)

d

faz a Soma de Riemann que calcula a área sob a curva em m intervalo bem definido como por exemplo:

[ a , b ] , ] -∞ , b ] ou [ a , ∞ [.

Este nome acima é dado em alusão ao matemático alemão Georg Friedrich Bernhard

Riemann (1826-1866).

A integral é um processo inverso do da derivada de funções pois,

( )

dt df f

( )

t C dt ) t ( df dt ) t ( dt df dt t f′ =

∫

=

∫

=

∫

= +

∫

ou

(

f

(

t

)

dt

)

f

(

t

)

dt

d

_⋅

₌

∫

. Mais precisamente:

∫

⋅

=

t a

f

(

t

)

dt

)

t

(

F

é chamada de primitiva de f(t).

Este resultado é chamado de Teorema Fundamental do Cálculo e faz a interligação entre o Cálculo Diferencial (secção anterior) e o Cálculo Integral (desta secção). Algumas regras de integração de funções em geral

( )

t dt a f

( )

t dt C f a = ⋅

∫

+

∫

(regra da homogeneidade)

( )

( )

[

f t +g t

]

dt =

∫

f

( )

t dt +

∫

g

( )

t dt + C

∫

(regra da aditividade)

( ) ( )

[

]

( )

_∫

( )

Se definirmos ) t ( g ) t ( u = e v(t) = f(t) então dt ) t ( g du = ′ ⋅ e dv=f′(t)⋅dt e a regra da integral por partes pode ser escrita doutra forma:

∫

∫

u⋅dv = uv − vdu Por outro lado, se

) t ( f ) t ( u = e du =f′(t)⋅dt, então a integral definida é calculada como:

]

u

(

b

)

u

(

a

)

u

du

b_a b a

=

=

−

∫

Fig. 1 – A área S sob a curva f(t) no intervalo definido [ a, b ].

A integral definida desde a até b da função f

S

d

)

(

f

b a

τ

⋅

τ

=

∫

A figura 2 mostra dois exemplos da integral definida desde a até b da função f, onde áreas abaixo do eixo das abcissas contam negativamente.

2 1 b a

f

1

(

τ

)

⋅

d

τ

=

S

−

S

∫

e 3 2 1 b a

f

2

(

τ

)

⋅

d

τ

=

S

−

S

+

S

∫

Fig. 2 – Dois exemplos da área sob a curva f(t) no intervalo definido [a, b]. As áreas abaixo do eixo das abcissas contam negativamente.

A figura 3 mostra dois exemplos da integral definida em intervalos infinitos: ] -∞, b ] e [ a , ∞ [.

'

S

d

)

(

f

b 3

τ

⋅

τ

=

∫

−∞ e

∫

a

f

4

(

τ

)

⋅

d

τ

=

S

''

∞ Fig. 3 – Dois exemplos da área sob a curva f(t) definidos em intervalos

Apresentamos agora uma tabela das integrais das principais funções. Integrais de funções racionais:

C u du = +

∫

1 n , C ) 1 n ( u du u 1 n n + ≠ + = ⋅ +

∫

C u ln u du du u 1⋅ =

∫

= +

∫

− C a u arctg a 1 dt a u 1 2 2 ⎟_⎠ + ⎞ ⎜ ⎝ ⎛ ⋅ = ⋅ +

∫

2 2 2 2 C, u >a a u a u arctg a 2 1 a u du ₊ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + − ⋅ = ⋅ −

∫

Integrais de funções irracionais:

C a u u ln a u du 2 2 2 2 + ⋅ = + + +

∫

C a u u ln a u du 2 2 2 2 − ⋅ = + − +

∫

C a u sec arc a 1 a u u du 2 2 ⎟⎠ + ⎞ ⎜ ⎝ ⎛ ⋅ = ⋅ − ⋅

∫

2 2 2 2 _a C , u <a u arcsen u a du ₊ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ⋅ −

∫

Integrais de logaritmos: C log t ) t a ( log t dt ) t a ( log_b ⋅ = ⋅ _b ⋅ − ⋅ _b +

∫

e (*) C t ) t a ( ln t dt ) t a ( ln ⋅ = ⋅ ⋅ − +

∫

[caso particular b = e da integral (*) acima]

(

n 1

)

C , n 1 t ) t a ( ln 1 n t dt ) t a ( ln t ₂ 1 n 1 n n + ≠ + − ⋅ ⋅ + = ⋅ ⋅ + +

∫

[

ln(a t)

]

C 2 1 dt ) t a ( ln t 1⋅ ⋅ = ⋅ ⋅ 2 +

∫

−

[

ln(a t)

]

C ln ) t a ( ln t dt _{= ⋅ +} ⋅ ⋅

∫

Integrais de funções exponenciais:

0 a , 1 a , C ) a ( ln a du a u u_{⋅ = + ≠ >}

∫

(**) C du u u⋅ = +

∫

e e [caso particular a = e da integral (**) acima]

C ) b ( ln b a 1 dt b at at = ⋅ +

∫

(***) C a 1 dt at at = +

∫

e e [caso particular b = e da integral (***) acima]

C ) 1 at ( a dt t ₂ at at = − + ⋅

∫

e e dt t a n t a 1 dt tn eat neat

∫

n 1eat

∫

_{⋅ = −} − 1 b , 0 b , dt b t ) b ln( a n ) b ln( a b t dt b t n 1 at at n at n > ≠ ⋅ − ⋅ = ⋅

∫

∫

−

(

)

[

a sen(bt) b cos(bt)

]

C b a dt ) t b ( sen _{2 2} at at ⋅ − ⋅ + + = ⋅

∫

e e

(

)

[

a cos(bt) b sen(bt)

]

C b a dt ) t b cos( _{2 2} at at ⋅ + ⋅ + + = ⋅

∫

e e

Integrais de funções trigonométricas:

( )

u du cos

( )

u C sen = − +

∫

( )

u du sen

( )

u C cos = +

∫

( )

u du ln

(

sec(u)

)

C tg = +

∫

( )

u du ln sen(u) C g cot = +

∫

( )

_{( )}

du ln sec(u) tg(u) C u cos 1 du u sec ⋅ =

∫

⋅ = + +

∫

( )

_{( )}

du ln cosec(u) cotg(u) C u sen 1 du u ec cos ⋅ =

∫

⋅ = − +

∫

( ) ( )

( )

_{( )}

du sec(u) C u sen u tg du u tg u ec s ⋅ ⋅ =

∫

⋅ = +

∫

( )

( )

_{( ) ( )}

du cosec(u) C u tg u sen 1 du u tg co u ec cos ⋅ = − + ⋅ = ⋅ ⋅

∫

∫

( )

_{( )}

du tg(u) C u cos 1 du u ec s 2 2 ⋅ = ⋅ = +

∫

∫

( )

_{( )}

du cotg(u) C u sen 1 du u ec cos 2 2 ⋅ = ⋅ = − +

∫

∫

( )

cos

( )

at C a 1 dt t a sen = − +

∫

( )

sen

( )

at C a 1 dt t a cos = +

∫

( )

( )

C a 4 t a 2 sen 2 t dt t a sen2 = − +

∫

( )

( )

C a 4 t a 2 sen 2 t dt t a cos2 = + +

∫

Fórmula de recorrência para integrais de potências de funções trigonométricas:

( )

_∫

( )

∫

+ − ⋅ ⋅ ⋅ ⋅ ⋅ − = ⋅ − − du u a sen n 1 n a n ) u a cos( ) u a ( sen du u a sen n 2 1 n n

( )

_∫

( )

∫

+ − ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = ⋅ − − du u a cos n 1 n a n ) u a ( sen ) u a ( cos du u a cos n 2 1 n n

( )

₍

₎

_∫

( )

∫

− ⋅ − ⋅ ⋅ = ⋅ − − du u a tg 1 n a ) u a ( tg du u a tg n 2 1 n n

( )

₍

₎

_∫

( )

∫

− ⋅ − ⋅ ⋅ − = ⋅ − − du u a g cot 1 n a ) u a ( g cot du u a g cot n 2 1 n n

( )

₍

₎

_∫

( )

∫

⎟⋅ ⋅ ⎠ ⎞ ⎜ ⎝ ⎛ − − + − ⋅ ⋅ ⋅ ⋅ = ⋅ − − du u a sec 1 n 2 n 1 n a ) u a ( tg ) u a ( sec du u a sec n 2 2 n n

( )

_∫

( )

∫

⋅ = −cosec − (a⋅u)⋅cotg(a⋅u) + ⎜⎛n−2⎞⎟⋅ n−2 ⋅ 2 n n

Integrais de outras funções trigonométricas:

( )

( )

[

]

[

]

2 2 b a , C ) b a ( 2 t ) b a ( cos ) b a ( 2 t ) b a ( cos dt t b cos t a sen + ≠ − − − + + − = ⋅ ⋅

∫

( )

( )

[

]

[

]

2 2 b a , C ) b a ( 2 t ) b a ( sen ) b a ( 2 t ) b a ( sen dt t b sen t a sen + ≠ + + − − − = ⋅ ⋅ ⋅

∫

( )

( )

[

]

[

]

2 2 b a , C ) b a ( 2 t ) b a ( sen ) b a ( 2 t ) b a ( sen dt t b cos t a cos + ≠ + + + − − = ⋅ ⋅ ⋅

∫

( )

( )

C a 4 ) t a 2 ( cos dt t a cos t a sen + ⋅ ⋅ ⋅ − = ⋅ ⋅ ⋅

∫

( )

( )

_{( )}

ln cos

( )

a t C a 1 dt t a cos t a sen dt t a tg = − ⋅ ⋅ + ⋅ ⋅ = ⋅

_∫

∫

( )

( )

_{( )}

ln sen

( )

at C a 1 dt t a sen t a cos dt t a g cot ⋅ =

∫

= ⋅ +

∫

( )

cos(a t) C a t ) t a ( sen a 1 dt t a sen t⋅ ⋅ = − ₂ ⋅ − ⋅ +

∫

( )

sin(at) C a t ) t a ( cos a 1 dt t a cos t⋅ = ₂ + +

∫

( )

t cos(at)dt a n ) t a ( cos a t dt t a sen t n1 n n

∫

∫

_{⋅ = − +} −

( )

t sen(at)dt a n ) t a ( sen a t dt t a cos t n 1 n n

∫

∫

_{⋅ = −} − Integrais de funções hiperbólicas:

C ) at ( cosh a 1 dt ) at ( senh = ⋅ +

∫

C ) at ( senh a 1 dt ) at ( cosh = ⋅ +

∫

C 2 t a 4 ) at 2 ( senh dt ) at ( senh2 = − +

∫

C 2 t a 4 ) at 2 ( senh dt ) at ( cosh2 = + +

∫

C ) at ( senh a 1 ) at ( cosh a t dt ) at ( senh t⋅ = ⋅ − ₂ ⋅ +

∫

C ) at ( cosh a 1 ) at ( senh a t dt ) at ( cosh t⋅ = ⋅ − ₂ ⋅ +

∫

C dt ) at ( cosh t a n ) at ( cosh a t dt ) at ( senh t n 1 n n _{⋅ = ⋅ − ⋅ ⋅ +}

∫

∫

− C dt ) at ( senh t a n ) at ( senh a t dt ) at ( cosh t n 1 n n⋅ = ⋅ − ⋅ ⋅ +

∫

∫

−

[

cosh(at)

]

C ln a 1 dt ) at ( cosh ) at ( senh dt ) at ( tanh =

∫

= ⋅ +

∫

C ) at ( senh ln a 1 dt ) at ( senh ) at ( cosh dt ) at ( coth =

∫

= ⋅ +

∫

Integrais definidas: π = ⋅

∫

∞ t dt 1₂ 0 -t e a 2 1 dt 0 ax2 = π

∫

∞ _e−

π

=

∫

∞ −

2

1

dt

0 t2

e

6 dt 1 t 2 0 π = ⋅ −

∫

∞ _et 15 dt 1 t 4 0 3 π = ⋅ −

∫

∞ t e 2 dt t ) t ( sen 0 π = ⋅

∫

∞

(

n 1

)

! ) n ( dt t 0 t 1 n ⋅ = Γ = −

∫

∞ − _e− [função gama]

( )

( )

⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ ≥ π ⋅ − ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ≥ π ⋅ ⋅ ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ ⋅ = ⋅ = ⋅

_∫

∫

π π 3 ímpar eiro int é n se , 2 ) 1 n ( 7 5 3 n 6 4 2 2 par eiro int é n se , 2 n 6 4 2 ) 1 n ( 5 2 1 dt t cos dt t sen 2 0 n 2 0 n L L L L