Lista de calculo 2

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FACULDADE DE CIÊNCIA E TECNOLOGIA CURSO ________________________________________________________________ DISCIPLINA: CÁLCULO APLICADO ALUNO (A): ______________________________________________________________ PROFESSOR(A)__________________________________________________________

Caro aluno, Pratique o máximo possível e não deixe de consultar o seu Professor nas possíveis dúvidas.

LISTA DEEXERCÍCIOS I

1) Use o conceito de primitiva e verifique se as seguintes integrais indefinidas estão corretas: (a)  tgx  dx   lncosx   c = ln|sec(x)| +C (c)  (e)  (g)
x2 e 2x dx  3  1  2 x 3  e c 6

(b)  cos(7x) dx  sen(7x)  c (d)  (cos 4 ) (f)
2

 2  sen 4 d 

2  sen4  6



3

c

3 dt  ln | lnt  | c t. lnt 

 1  x 2 dx  2arctgx   csen(3t ) 1  ln | 1  cos(3t ) |  C 1  cos(3t ) 3



e

y

dy  e

y

c

y

(h) 

2) Determine: a) Uma função f(x) tal que f ‘ (x) + 6 sen(3x) = 0 e f (0) = 5 (2x 2 - 1) 2 b) A primitiva F(x) da função f (x) = que passa pelo ponto P=(1, 3/2) x3 1 c) A imagem f   , sabendo-se que  f( x)dx  sen x  x. cos x  x 2  C 4 2
1

3) Resolva as integrais abaixo usandosubstituição de variável:

a)  25x dx b)  sen(ax )dx , com a  0 c) 

dx sen (3x  1)
2

25x Re sp. : C 5 ln( 2) cos(ax ) Re sp. :  C a cot g(3x  1) Re sp. :  C 3
Re sp. : sen( 5x ) C 5

d)  cos(5x )dx e) 
dx 3x  7

1 Re sp. : ln 3x  7  C 3 1 Re sp. :  ln cos 2x  C 2

f)  tg(2x )dx g)  (cot g(e x )e x dx h)  x 2  1.xdx i)  j)  l)

Re sp. : ln sen(ex )  C
Re sp.:
Re sp. :

1 ( x 2  1) 3  C 3
1 2x 2  3 2 C

xdx 2x 2  3

cot g( x ) sen 2 x

dx

cot g 2 x Re sp. :  C 2

 cos 2 x

dx tgx 1

Re sp. : 2 tgx  1  C
ln 2 ( x  1) C 2

m)  n)  n) 

ln( x  1) dx x 1

Re sp. :

cos xdx 2 sen x 1
sen(2x )dx 1  sen 2 x

Re sp. : 2 sen x  1  C

Re sp. : 2 1  sen 2 x  C

2

o) 

arcsen xdx 1  x2 arctg 2xdx 1 x2

arcsen 2 x Re sp. : C 2

p)  q)  r) 

Re sp. :

arctg 3 x C 3

dx x 2  2x  3
dx x ln x
2

x 1

1 Re sp. : ln x 2  2x  3  C 2

Re sp. : ln ln x  C
( x  2)dx

s)  3x t) 

 4 x 3

dx 1  2x 2
dx 16  9x 2

3x  4 x 3 Re sp. C 2. ln(3) 1 Re sp. arctg( 2.x )  C 2
1 3x Re sp. arcsen C 3 4

2

u) 

4) Use integração por partes pararesolver as integrais: Resp.: x2 ex + C Resp.: ln(x).(4x4+2x2+x) - (x4+x2 + x) + C Resp.: - (x2 –1) cos(x) +2xsen(x) + C
1 Re sp. : x.arctg(3x )  ln(9x 2  1)  C 6

a)  ( x 2  2x )e x dx b)  (16x 3  4x  1) ln( x)dx c)  ( x 2  1) sen xdx d)  arctg(3x )dx
e) arcsen(x  2)dx



Re sp. : ( x  2) arcsen( x  2)   x 2  4x  3  C

3

f)




x dx sen 2 x
3x8.cos(x3 )dx

Resp. : x cot g(x)  ln | sen(x) | C
Re sp. : x 6 sen(x3 )  2x3 cos(x3 )  2 sen(x3 )  C

g)

3 h) x5 (1  4e x )dx



Re sp. : e

x 3  4x 

3

 

 4  x6  C 3  6 
2 x 1

i)  e 2x 1 .dx

Re sp. : ( 2x  1  1)e

C

5) Resolva as integrais contendo um trinômio ax2 + bx + c:

a)



dx x  2x  5
2

1 x 1 Re sp. : arctg C 2 2

b)



dx x 6x  5
2

1 x 5 Re sp. : ln C 4 x 1
1 Re sp. : ln | 2x 2  4x  3 | 2 2.arctg[ 2 ( x  1)]  C 4

c)




(x  5)dx 2x 2  4x  3
x3 3  4x  4x 2 dx

d)

Re sp. : 

1 7 2x  1 3  4x  4x 2  arcsen C 4 4 2

4

6) Resolva as integrais de funções racionais:

a)



x 1 dx 2x 1
xdx ( x  1)( x  3)( x  5) dx ( x  1)2 ( x  2)

1 1 Re sp. x  ln 2x  1 C 2 4

b)

 

1 ( x  3)6 Re sp. ln C 8 ( x  5)5 ( x  1)
Re sp. 1 x2  ln C x 1 x 1
2

c)

d)



x 8 dx x  4x2  4x
3

3  x 2 Re sp.  ln   C x2  x 
x 1 Re sp. :  ln | x | [9 ln | 2x  1| 7 ln | 2x  1 |]  C 4 16
( x  2x x 1
2 3  5) 2

e)

 2 x  3x  3 f)  (x 1)(x  2x  5)dx
2 2

x3  1 dx 4x3  x

Re sp. : ln

1  x 1  arctg...
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