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  Universidade de AveiroDepartamento de Matem´atica C´ALCULO II - Agrupamento I - 2015/16Formul´ario (Transformada de Laplace) fun¸c˜ao transformada t n ( n  ∈ N 0 )  n ! s n +1  , s >  0e at ( a  ∈ R ) 1 s − a , s > a sen( at ) ( a  ∈ R )  as 2 +  a 2  , s >  0cos( at ) ( a  ∈ R )  ss 2 +  a 2  , s >  0senh( at ) ( a  ∈ R )  as 2 − a 2  , s >  | a | cosh( at ) ( a  ∈ R )  ss 2 − a 2  , s >  | a | f  ( t ) +  g ( t )  F  ( s ) +  G ( s ) , s > s f  ,s g αf  ( t ) ( α  ∈ R )  αF  ( s ) , s > s f  e λt f  ( t ) ( λ  ∈ R )  F  ( s − λ ) , s > s f   +  λt n f  ( t ) ( n  ∈ N ) ( − 1) n F  ( n ) ( s ) , s > s f  f  ( t − a ) ( a >  0) e − as F  ( s ) , s > s f  f  ( at ) ( a >  0) 1 a F   sa   , s > as f  f  ( n ) ( t ) ( n  ∈ N )  s n F  ( s ) − n  k =1 s n − k f  ( k − 1) (0) ,  onde  f  (0) ≡  f ,s >  max { s f  ,s f   ,s f    ...,s f  ( n − 1) } Notas: 1.  F   denota a transformada de Laplace da fun¸c˜ao  f  ,  F  ( s ) =  L{ f  ( t ) } ( s );2. O facto de se indicarem restri¸c˜oes numa dada linha do quadro acima n˜ao significa que n˜ao haja restri¸c˜oes adicionais a considerar para que a f´ormula indicada nessa linha seja v´alida.  Formul´ario (Primitivas) Fun¸c˜ao Primitiva Fun¸c˜ao Primitiva u r u  , com  r   =  − 1  u r +1 r  + 1 u  u  ln | u | u  e u e u u  a u  a u ln au  cos u  sen u u  sen u  − cos uu  sec 2 u  tg u u  cosec 2 u  − cotg uu  sec u  ln | sec u  + tg u |  u  cosec u  − ln | cosec u  + cotg u | u  √  1 − u 2 − arccos u ouarcsen uu  1 +  u 2 arctan u ou − arccotg u Algumas f´ormulas trigonom´etricas: sec u  = 1cos u  cosec u  = 1sen u  sen 2 u  + cos 2 u  = 1cos 2 u  = 1 + cos2 u 2 sen 2 u  = 1 − cos2 u 2 1 + tg 2 u  = sec 2 u 1 + cotg 2 u  = cosec 2 u  sen( u  +  v ) = sen u cos v  + sen v  cos u  cos( u  +  v ) = cos u cos v − sen u sen v 2
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