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Revista Electronica MateInfo.RO ISSN 2065-6432 MAI 2010

 

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                O metoda de calcul al primitivelor

 

                                  de Marian Teler, Profesor Costesti, Judetul Arges

                                si Marin Ionescu, Profesor Pitesti, Judetul Arges

 

            In manualul de analiza matematica pentru clasa a XII-a, capitolul „ MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=5biaaa@37CB@ Integrarea prin parti”, sunt propuse o serie de exercitii care necesita aplicarea de doua sau mai multe ori a formulei de integrare prin parti.

            Vazand rezultatele, observam ca la anumite tipuri de functii se obtin anumite tipuri de primitive.

Propunem in continuare rezolvarea unor probleme de calcul al primitivelor prin metoda coeficientilor nedeterminati.

 

  1. Fie f:R→R,f(x)=P(x) e x ,P∈R[X] MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdacaWGsbGaeyOKH4QaamOuaiaacYcacaWGMbGaaiikaiaadIhacaGGPaGaeyypa0JaamiuaiaacIcacaWG4bGaaiykaiaadwgadaahaaWcbeqaaiaadIhaaaGccaGGSaGaamiuaiabgIGiolaadkfacaGGBbGaamiwaiaac2faaaa@4BF4@ .

      Exista o primitiva F:R→R,f(x)=Q(x) e x MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaacQdacaWGsbGaeyOKH4QaamOuaiaacYcacaWGMbGaaiikaiaadIhacaGGPaGaeyypa0JaamyuaiaacIcacaWG4bGaaiykaiaadwgadaahaaWcbeqaaiaadIhaaaaaaa@454E@ , unde Q∈R[X] MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuaiabgIGiolaadkfacaGGBbGaamiwaiaac2faaaa@3BC1@ , grad(Q)=grad(P) MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiaadkhacaWGHbGaamizaiaacIcacaWGrbGaaiykaiabg2da9iaadEgacaWGYbGaamyyaiaadsgacaGGOaGaamiuaiaacMcaaaa@42BA@

      Din relatia F ' =f MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaCaaaleqabaGaai4jaaaakiabg2da9iaadAgaaaa@3991@ , obtinem Q ' +Q=P MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuamaaCaaaleqabaGaai4jaaaakiabgUcaRiaadgfacqGH9aqpcaWGqbaaaa@3B3E@

Exemple:

1.a.  Fie f:R→R,f(x)=( x 3 −x+1) e x MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdacaWGsbGaeyOKH4QaamOuaiaacYcacaWGMbGaaiikaiaadIhacaGGPaGaeyypa0JaaiikaiaadIhadaahaaWcbeqaaiaaiodaaaGccqGHsislcaWG4bGaey4kaSIaaGymaiaacMcacaWGLbWaaWbaaSqabeaacaWG4baaaaaa@4913@ . Sa se determine a,b,c∈R MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacYcacaWGIbGaaiilaiaadogacqGHiiIZcaWGsbaaaa@3C63@  astfel incat 

 functia F:R→R,F(x)=( x 3 +a x 2 +bx+c) e x MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaacQdacaWGsbGaeyOKH4QaamOuaiaacYcacaWGgbGaaiikaiaadIhacaGGPaGaeyypa0JaaiikaiaadIhadaahaaWcbeqaaiaaiodaaaGccqGHRaWkcaWGHbGaamiEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadkgacaWG4bGaey4kaSIaam4yaiaacMcacaWGLbWaaWbaaSqabeaacaWG4baaaaaa@4D94@  sa fie o primitiva a functiei f.

 Solutie:

 Din Q ' +Q=P MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuamaaCaaaleqabaGaai4jaaaakiabgUcaRiaadgfacqGH9aqpcaWGqbaaaa@3B3E@ , rezulta x 3 +(a+3) x 2 +(b+2a)x+b+c= x 3 −x+1,( ∀ )x∈R MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCaaaleqabaGaaG4maaaakiabgUcaRiaacIcacaWGHbGaey4kaSIaaG4maiaacMcacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaiikaiaadkgacqGHRaWkcaaIYaGaamyyaiaacMcacaWG4bGaey4kaSIaamOyaiabgUcaRiaadogacqGH9aqpcaWG4bWaaWbaaSqabeaacaaIZaaaaOGaeyOeI0IaamiEaiabgUcaRiaaigdacaGGSaWaaeWaaeaacqGHaiIiaiaawIcacaGLPaaacaWG4bGaeyicI4SaamOuaaaa@55A9@

 Identificand coeficientii, obtinem sistemul:

{ a+3=0 b+2a=−1 b+c=1 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaqaabeqaaiaadggacqGHRaWkcaaIZaGaeyypa0JaaGimaaqaaiaadkgacqGHRaWkcaaIYaGaamyyaiabg2da9iabgkHiTiaaigdaaeaacaWGIbGaey4kaSIaam4yaiabg2da9iaaigdaaaGaay5Eaaaaaa@45E5@   ,  { a=−3 b=5 c=−4 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaqaabeqaaiaadggacqGH9aqpcqGHsislcaaIZaaabaGaamOyaiabg2da9iaaiwdaaeaacaWGJbGaeyypa0JaeyOeI0IaaGinaaaacaGL7baaaaa@40F0@  si  F(x)=( x 3 −3 x 2 +5x−4 ) e x MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaacIcacaWG4bGaaiykaiabg2da9maabmaabaGaamiEamaaCaaaleqabaGaaG4maaaakiabgkHiTiaaiodacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGynaiaadIhacqGHsislcaaI0aaacaGLOaGaayzkaaGaamyzamaaCaaaleqabaGaamiEaaaaaaa@478B@

1.b. Generalizare:

Fie f:R→R,f(x)=( x n + a 1 x n−1 +...+ a n ) e x MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdacaWGsbGaeyOKH4QaamOuaiaacYcacaWGMbGaaiikaiaadIhacaGGPaGaeyypa0JaaiikaiaadIhadaahaaWcbeqaaiaad6gaaaGccqGHRaWkcaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaamiEamaaCaaaleqabaGaamOBaiabgkHiTiaaigdaaaGccqGHRaWkcaGGUaGaaiOlaiaac6cacqGHRaWkcaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaaiykaiaadwgadaahaaWcbeqaaiaadIhaaaaaaa@5233@ , a 1 , a 2 ,... a n ∈R MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaaIXaaabeaakiaacYcacaWGHbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaac6cacaGGUaGaaiOlaiaadggadaWgaaWcbaGaamOBaaqabaGccqGHiiIZcaWGsbaaaa@4182@ .

Sa se determine  b 1 , b 2 ,... b n ∈R MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaaIXaaabeaakiaacYcacaWGIbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaac6cacaGGUaGaaiOlaiaadkgadaWgaaWcbaGaamOBaaqabaGccqGHiiIZcaWGsbaaaa@4185@ , astfel incat functia F:R→R,F(x)=( x n + b 1 x n−1 +...+ b n ) e x MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaacQdacaWGsbGaeyOKH4QaamOuaiaacYcacaWGgbGaaiikaiaadIhacaGGPaGaeyypa0JaaiikaiaadIhadaahaaWcbeqaaiaad6gaaaGccqGHRaWkcaWGIbWaaSbaaSqaaiaaigdaaeqaaOGaamiEamaaCaaaleqabaGaamOBaiabgkHiTiaaigdaaaGccqGHRaWkcaGGUaGaaiOlaiaac6cacqGHRaWkcaWGIbWaaSbaaSqaaiaad6gaaeqaaOGaaiykaiaadwgadaahaaWcbeqaaiaadIhaaaaaaa@51F5@  sa fie o primitiva a functiei f.

 Solutie:

 Luand b 0 =1 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaaIWaaabeaakiabg2da9iaaigdaaaa@398B@ , din Q ' +Q=P MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyuamaaCaaaleqabaGaai4jaaaakiabgUcaRiaadgfacqGH9aqpcaWGqbaaaa@3B3E@ , obtinem b k = a k −( n−k+1 ) b k−1 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaWGRbaabeaakiabg2da9iaadggadaWgaaWcbaGaam4AaaqabaGccqGHsisldaqadaqaaiaad6gacqGHsislcaWGRbGaey4kaSIaaGymaaGaayjkaiaawMcaaiaadkgadaWgaaWcbaGaam4AaiabgkHiTiaaigdaaeqaaaaa@45A0@ , k∈{ 1,2,...,n } MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgIGiopaacmaabaGaaGymaiaacYcacaaIYaGaaiilaiaac6cacaGGUaGaaiOlaiaacYcacaWGUbaacaGL7bGaayzFaaaaaa@4128@

  1. Fie f:( 0,∞ )→R,f(x)= x n ln k x MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdadaqadaqaaiaaicdacaGGSaGaeyOhIukacaGLOaGaayzkaaGaeyOKH4QaamOuaiaacYcacaWGMbGaaiikaiaadIhacaGGPaGaeyypa0JaamiEamaaCaaaleqabaGaamOBaaaakiGacYgacaGGUbWaaWbaaSqabeaacaWGRbaaaOGaamiEaaaa@49EA@ ,  n,k∈ N ∗ MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacYcacaWGRbGaeyicI4SaamOtamaaCaaaleqabaGaey4fIOcaaaaa@3BF9@ . Sa se determine  b 1 , b 2 ,... b k ∈R MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaaIXaaabeaakiaacYcacaWGIbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaac6cacaGGUaGaaiOlaiaadkgadaWgaaWcbaGaam4AaaqabaGccqGHiiIZcaWGsbaaaa@4182@ , astfel incat functia F:( 0,∞ )→R,F(x)= x n+1 n+1 ( ln k x+ b 1 ln k−1 x+...+ b k ) MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaacQdadaqadaqaaiaaicdacaGGSaGaeyOhIukacaGLOaGaayzkaaGaeyOKH4QaamOuaiaacYcacaWGgbGaaiikaiaadIhacaGGPaGaeyypa0ZaaSaaaeaacaWG4bWaaWbaaSqabeaacaWGUbGaey4kaSIaaGymaaaaaOqaaiaad6gacqGHRaWkcaaIXaaaaiaacIcaciGGSbGaaiOBamaaCaaaleqabaGaam4AaaaakiaadIhacqGHRaWkcaWGIbWaaSbaaSqaaiaaigdaaeqaaOGaciiBaiaac6gadaahaaWcbeqaaiaadUgacqGHsislcaaIXaaaaOGaamiEaiabgUcaRiaac6cacaGGUaGaaiOlaiabgUcaRiaadkgadaWgaaWcbaGaam4AaaqabaGccaGGPaaaaa@5D91@  sa fie o primitiva a functiei f.

Solutie:

     Din F ' (x)=f(x),( ∀ )x∈( 0,∞ ) MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaCaaaleqabaGaai4jaaaakiaacIcacaWG4bGaaiykaiabg2da9iaadAgacaGGOaGaamiEaiaacMcacaGGSaWaaeWaaeaacqGHaiIiaiaawIcacaGLPaaacaWG4bGaeyicI48aaeWaaeaacaaIWaGaaiilaiabg6HiLcGaayjkaiaawMcaaaaa@482B@ , obtinem (considerand b 0 =1 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaaIWaaabeaakiabg2da9iaaigdaaaa@398B@  ), 

             b i = k−i+1 n+1 ,i∈{ 1,2,...,k } MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaWGPbaabeaakiabg2da9maalaaabaGaam4AaiabgkHiTiaadMgacqGHRaWkcaaIXaaabaGaamOBaiabgUcaRiaaigdaaaGaaiilaiaadMgacqGHiiIZdaGadaqaaiaaigdacaGGSaGaaGOmaiaacYcacaGGUaGaaiOlaiaac6cacaGGSaGaam4AaaGaay5Eaiaaw2haaaaa@4BEC@

Exemplu:  Fie f:( 0,∞ )→R,f(x)= x 3 ln 2 x MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdadaqadaqaaiaaicdacaGGSaGaeyOhIukacaGLOaGaayzkaaGaeyOKH4QaamOuaiaacYcacaWGMbGaaiikaiaadIhacaGGPaGaeyypa0JaamiEamaaCaaaleqabaGaaG4maaaakiGacYgacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaamiEaaaa@4980@ . Sa se determine a,b∈R MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacYcacaWGIbGaeyicI4SaamOuaaaa@3ACB@  astfel incat 

functia F:( 0,∞ )→R,F(x)= x 4 4 ( ln 2 x+alnx+b) MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaacQdadaqadaqaaiaaicdacaGGSaGaeyOhIukacaGLOaGaayzkaaGaeyOKH4QaamOuaiaacYcacaWGgbGaaiikaiaadIhacaGGPaGaeyypa0ZaaSaaaeaacaWG4bWaaWbaaSqabeaacaaI0aaaaaGcbaGaaGinaaaacaGGOaGaciiBaiaac6gadaahaaWcbeqaaiaaikdaaaGccaWG4bGaey4kaSIaamyyaiGacYgacaGGUbGaamiEaiabgUcaRiaadkgacaGGPaaaaa@51DA@  sa fie o primitiva a functiei f.

Din F ' (x)=f(x),( ∀ )x∈( 0,∞ ) MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaCaaaleqabaGaai4jaaaakiaacIcacaWG4bGaaiykaiabg2da9iaadAgacaGGOaGaamiEaiaacMcacaGGSaWaaeWaaeaacqGHaiIiaiaawIcacaGLPaaacaWG4bGaeyicI48aaeWaaeaacaaIWaGaaiilaiabg6HiLcGaayjkaiaawMcaaaaa@482B@ , obtinem a=− 1 2 ,b= 1 8 ,F(x)= x 4 4 ( ln 2 x− 1 2 x+ 1 8 ) MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2da9iabgkHiTmaalaaabaGaaGymaaqaaiaaikdaaaGaaiilaiaadkgacqGH9aqpdaWcaaqaaiaaigdaaeaacaaI4aaaaiaacYcacaWGgbGaaiikaiaadIhacaGGPaGaeyypa0ZaaSaaaeaacaWG4bWaaWbaaSqabeaacaaI0aaaaaGcbaGaaGinaaaadaqadaqaaiGacYgacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaamiEaiabgkHiTmaalaaabaGaaGymaaqaaiaaikdaaaGaamiEaiabgUcaRmaalaaabaGaaGymaaqaaiaaiIdaaaaacaGLOaGaayzkaaaaaa@5151@

 

  1. Fie f:R→R,f(x)=(asinx+bcosx) e x ,a,b∈R MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdacaWGsbGaeyOKH4QaamOuaiaacYcacaWGMbGaaiikaiaadIhacaGGPaGaeyypa0JaaiikaiaadggaciGGZbGaaiyAaiaac6gacaWG4bGaey4kaSIaamOyaiGacogacaGGVbGaai4CaiaadIhacaGGPaGaamyzamaaCaaaleqabaGaamiEaaaakiaacYcacaWGHbGaaiilaiaadkgacqGHiiIZcaWGsbaaaa@5381@ . 

     Sa se determine m,n∈R MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiaacYcacaWGUbGaeyicI4SaamOuaaaa@3AE3@ , astfel incat functia  F:R→R,F(x)=(msinx+ncosx) e x MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaacQdacaWGsbGaeyOKH4QaamOuaiaacYcacaWGgbGaaiikaiaadIhacaGGPaGaeyypa0Jaaiikaiaad2gaciGGZbGaaiyAaiaac6gacaWG4bGaey4kaSIaamOBaiGacogacaGGVbGaai4CaiaadIhacaGGPaGaamyzamaaCaaaleqabaGaamiEaaaaaaa@4DC7@

Solutie:

Din F ' (x)=f(x),( ∀ )x∈R MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaCaaaleqabaGaai4jaaaakiaacIcacaWG4bGaaiykaiabg2da9iaadAgacaGGOaGaamiEaiaacMcacaGGSaWaaeWaaeaacqGHaiIiaiaawIcacaGLPaaacaWG4bGaeyicI4SaamOuaaaa@449E@ , obtinem: { m−n=a m+n=b MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaqaabeqaaiaad2gacqGHsislcaWGUbGaeyypa0Jaamyyaaqaaiaad2gacqGHRaWkcaWGUbGaeyypa0JaamOyaaaacaGL7baaaaa@4086@  ,  { m= a+b 2 n= −a+b 2 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaqaabeqaaiaad2gacqGH9aqpdaWcaaqaaiaadggacqGHRaWkcaWGIbaabaGaaGOmaaaaaeaacaWGUbGaeyypa0ZaaSaaaeaacqGHsislcaWGHbGaey4kaSIaamOyaaqaaiaaikdaaaaaaiaawUhaaaaa@42E8@

Exemplu:

Fie f:R→R,f(x)=(sinx−cosx) e x MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdacaWGsbGaeyOKH4QaamOuaiaacYcacaWGMbGaaiikaiaadIhacaGGPaGaeyypa0JaaiikaiGacohacaGGPbGaaiOBaiaadIhacqGHsislciGGJbGaai4BaiaacohacaWG4bGaaiykaiaadwgadaahaaWcbeqaaiaadIhaaaaaaa@4C2D@ . 

Sa se determine m,n∈R MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiaacYcacaWGUbGaeyicI4SaamOuaaaa@3AE3@ , astfel incat functia  F:R→R,F(x)=(msinx+ncosx) e x MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaacQdacaWGsbGaeyOKH4QaamOuaiaacYcacaWGgbGaaiikaiaadIhacaGGPaGaeyypa0Jaaiikaiaad2gaciGGZbGaaiyAaiaac6gacaWG4bGaey4kaSIaamOBaiGacogacaGGVbGaai4CaiaadIhacaGGPaGaamyzamaaCaaaleqabaGaamiEaaaaaaa@4DC7@  sa fie o primitiva a functiei f.

Solutie: Avem a=1,b=−1 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabg2da9iaaigdacaGGSaGaamOyaiabg2da9iabgkHiTiaaigdaaaa@3CDF@ , rezulta m=0,n=−1 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabg2da9iaaicdacaGGSaGaamOBaiabg2da9iabgkHiTiaaigdaaaa@3CF6@ ,  F(x)=(−cosx) e x MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaacIcacaWG4bGaaiykaiabg2da9iaacIcacqGHsislciGGJbGaai4BaiaacohacaWG4bGaaiykaiaadwgadaahaaWcbeqaaiaadIhaaaaaaa@4244@

  1. Fie f:R→R,f(x)=P(x)sinx+Q(x)cosx,P,Q∈R[X] MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdacaWGsbGaeyOKH4QaamOuaiaacYcacaWGMbGaaiikaiaadIhacaGGPaGaeyypa0JaamiuaiaacIcacaWG4bGaaiykaiGacohacaGGPbGaaiOBaiaadIhacqGHRaWkcaWGrbGaaiikaiaadIhacaGGPaGaci4yaiaac+gacaGGZbGaamiEaiaacYcacaWGqbGaaiilaiaadgfacqGHiiIZcaWGsbGaai4waiaadIfacaGGDbaaaa@570F@ .

 Exista R,S∈R[X] MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiaacYcacaWGtbGaeyicI4SaamOuaiaacUfacaWGybGaaiyxaaaa@3D4A@ ,   max( gradR,gradS )≤max( gradP,gradQ ) MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyBaiaacggacaGG4bWaaeWaaeaacaWGNbGaamOCaiaadggacaWGKbGaamOuaiaacYcacaWGNbGaamOCaiaadggacaWGKbGaam4uaaGaayjkaiaawMcaaiabgsMiJkGac2gacaGGHbGaaiiEamaabmaabaGaam4zaiaadkhacaWGHbGaamizaiaadcfacaGGSaGaam4zaiaadkhacaWGHbGaamizaiaadgfaaiaawIcacaGLPaaaaaa@53E4@ , astfel incat functia F:R→R,F(x)=R(x)sinx+S(x)cosx MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaacQdacaWGsbGaeyOKH4QaamOuaiaacYcacaWGgbGaaiikaiaadIhacaGGPaGaeyypa0JaamOuaiaacIcacaWG4bGaaiykaiGacohacaGGPbGaaiOBaiaadIhacqGHRaWkcaWGtbGaaiikaiaadIhacaGGPaGaci4yaiaac+gacaGGZbGaamiEaaaa@4ED0@  sa fie o primitiva a functiei f.

 Solutie: Din F ' (x)=f(x),( ∀ )x∈R MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaCaaaleqabaGaai4jaaaakiaacIcacaWG4bGaaiykaiabg2da9iaadAgacaGGOaGaamiEaiaacMcacaGGSaWaaeWaaeaacqGHaiIiaiaawIcacaGLPaaacaWG4bGaeyicI4SaamOuaaaa@449E@ , obtinem: { R ' −S=P R+ S ' =Q MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaqaabeqaaiaadkfadaahaaWcbeqaaiaacEcaaaGccqGHsislcaWGtbGaeyypa0JaamiuaaqaaiaadkfacqGHRaWkcaWGtbWaaWbaaSqabeaacaGGNaaaaOGaeyypa0JaamyuaaaacaGL7baaaaa@41BC@

Exemplu: Fie f:R→R,f(x)=( x 2 −x+1)sinx MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdacaWGsbGaeyOKH4QaamOuaiaacYcacaWGMbGaaiikaiaadIhacaGGPaGaeyypa0JaaiikaiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWG4bGaey4kaSIaaGymaiaacMcaciGGZbGaaiyAaiaac6gacaWG4baaaa@4AD3@ . Sa se determine a,b,c,m,n,p∈R MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacYcacaWGIbGaaiilaiaadogacaGGSaGaamyBaiaacYcacaWGUbGaaiilaiaadchacqGHiiIZcaWGsbaaaa@414D@ , astfel incat functia F:R→R,F(x)=(a x 2 +bx+c)sinx+(m x 2 +nx+p)cosx MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaacQdacaWGsbGaeyOKH4QaamOuaiaacYcacaWGgbGaaiikaiaadIhacaGGPaGaeyypa0JaaiikaiaadggacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamOyaiaadIhacqGHRaWkcaWGJbGaaiykaiGacohacaGGPbGaaiOBaiaadIhacqGHRaWkcaGGOaGaamyBaiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGUbGaamiEaiabgUcaRiaadchacaGGPaGaci4yaiaac+gacaGGZbGaamiEaaaa@5A18@  sa fie o primitiva a functiei f.

Solutie: Din F ' (x)=f(x),( ∀ )x∈R MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaCaaaleqabaGaai4jaaaakiaacIcacaWG4bGaaiykaiabg2da9iaadAgacaGGOaGaamiEaiaacMcacaGGSaWaaeWaaeaacqGHaiIiaiaawIcacaGLPaaacaWG4bGaeyicI4SaamOuaaaa@449E@ , obtinem: { m=1 b+2m=0 b−p=1 a=0 2a−n=1 c+n=0 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaqaabeqaaiaad2gacqGH9aqpcaaIXaaabaGaamOyaiabgUcaRiaaikdacaWGTbGaeyypa0JaaGimaaqaaiaadkgacqGHsislcaWGWbGaeyypa0JaaGymaaqaaiaadggacqGH9aqpcaaIWaaabaGaaGOmaiaadggacqGHsislcaWGUbGaeyypa0JaaGymaaqaaiaadogacqGHRaWkcaWGUbGaeyypa0JaaGimaaaacaGL7baaaaa@4FF2@  ,  { a=0 b=2 c=−1 m=−1 n=1 p=1 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaqaabeqaaiaadggacqGH9aqpcaaIWaaabaGaamOyaiabg2da9iaaikdaaeaacaWGJbGaeyypa0JaeyOeI0IaaGymaaqaaiaad2gacqGH9aqpcqGHsislcaaIXaaabaGaamOBaiabg2da9iaaigdaaeaacaWGWbGaeyypa0JaaGymaaaacaGL7baaaaa@4907@    

           F(x)=( 2x−1 )sinx+( − x 2 +x+1 )cosx MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaacIcacaWG4bGaaiykaiabg2da9maabmaabaGaaGOmaiaadIhacqGHsislcaaIXaaacaGLOaGaayzkaaGaci4CaiaacMgacaGGUbGaamiEaiabgUcaRmaabmaabaGaeyOeI0IaamiEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadIhacqGHRaWkcaaIXaaacaGLOaGaayzkaaGaci4yaiaac+gacaGGZbGaamiEaaaa@4F6D@

  1. Fie f:R→R,f(x)= e x [ P(x)sinx+Q(x)cosx ],P,Q∈R[X] MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdacaWGsbGaeyOKH4QaamOuaiaacYcacaWGMbGaaiikaiaadIhacaGGPaGaeyypa0JaamyzamaaCaaaleqabaGaamiEaaaakmaadmaabaGaamiuaiaacIcacaWG4bGaaiykaiGacohacaGGPbGaaiOBaiaadIhacqGHRaWkcaWGrbGaaiikaiaadIhacaGGPaGaci4yaiaac+gacaGGZbGaamiEaaGaay5waiaaw2faaiaacYcacaWGqbGaaiilaiaadgfacqGHiiIZcaWGsbGaai4waiaadIfacaGGDbaaaa@5B1F@ . Exista R,S∈R[X] MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiaacYcacaWGtbGaeyicI4SaamOuaiaacUfacaWGybGaaiyxaaaa@3D4A@ , max( gradR,gradS )≤max( gradP,gradQ ) MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyBaiaacggacaGG4bWaaeWaaeaacaWGNbGaamOCaiaadggacaWGKbGaamOuaiaacYcacaWGNbGaamOCaiaadggacaWGKbGaam4uaaGaayjkaiaawMcaaiabgsMiJkGac2gacaGGHbGaaiiEamaabmaabaGaam4zaiaadkhacaWGHbGaamizaiaadcfacaGGSaGaam4zaiaadkhacaWGHbGaamizaiaadgfaaiaawIcacaGLPaaaaaa@53E4@ , astfel incat F:R→R,F(x)= e x [ R(x)sinx+S(x)cosx ] MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaacQdacaWGsbGaeyOKH4QaamOuaiaacYcacaWGgbGaaiikaiaadIhacaGGPaGaeyypa0JaamyzamaaCaaaleqabaGaamiEaaaakmaadmaabaGaamOuaiaacIcacaWG4bGaaiykaiGacohacaGGPbGaaiOBaiaadIhacqGHRaWkcaWGtbGaaiikaiaadIhacaGGPaGaci4yaiaac+gacaGGZbGaamiEaaGaay5waiaaw2faaaaa@52E0@  sa fie o primitiva a functiei f.

      Exemplu: Fie f:R→R,f(x)=x e x sinx MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdacaWGsbGaeyOKH4QaamOuaiaacYcacaWGMbGaaiikaiaadIhacaGGPaGaeyypa0JaamiEaiaadwgadaahaaWcbeqaaiaadIhaaaGcciGGZbGaaiyAaiaac6gacaWG4baaaa@471E@ . Sa se determine a,b,m,n∈R MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaacYcacaWGIbGaaiilaiaad2gacaGGSaGaamOBaiabgIGiolaadkfaaaa@3E10@  astfel incat        

      functia F:R→R,F(x)= e x [ ( ax+b )sinx+( mx+n )cosx ] MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaacQdacaWGsbGaeyOKH4QaamOuaiaacYcacaWGgbGaaiikaiaadIhacaGGPaGaeyypa0JaamyzamaaCaaaleqabaGaamiEaaaakmaadmaabaWaaeWaaeaacaWGHbGaamiEaiabgUcaRiaadkgaaiaawIcacaGLPaaaciGGZbGaaiyAaiaac6gacaWG4bGaey4kaSYaaeWaaeaacaWGTbGaamiEaiabgUcaRiaad6gaaiaawIcacaGLPaaaciGGJbGaai4BaiaacohacaWG4baacaGLBbGaayzxaaaaaa@5707@  sa fie o primitiva a functiei f.

      Solutie: Din F ' (x)=f(x),( ∀ )x∈R MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaCaaaleqabaGaai4jaaaakiaacIcacaWG4bGaaiykaiabg2da9iaadAgacaGGOaGaamiEaiaacMcacaGGSaWaaeWaaeaacqGHaiIiaiaawIcacaGLPaaacaWG4bGaeyicI4SaamOuaaaa@449E@ , obtinem: { a−m=1 a+m=0 b+n+m=0 b−n+a=0 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaqaabeqaaiaadggacqGHsislcaWGTbGaeyypa0JaaGymaaqaaiaadggacqGHRaWkcaWGTbGaeyypa0JaaGimaaqaaiaadkgacqGHRaWkcaWGUbGaey4kaSIaamyBaiabg2da9iaaicdaaeaacaWGIbGaeyOeI0IaamOBaiabgUcaRiaadggacqGH9aqpcaaIWaaaaiaawUhaaaaa@4CB5@  , { a= 1 2 b=0 m=− 1 2` n= 1 2 MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiqaaqaabeqaaiaadggacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaaaqaaiaadkgacqGH9aqpcaaIWaaabaGaamyBaiabg2da9iabgkHiTmaalaaabaGaaGymaaqaaiaaikdacaGGGbaaaaqaaiaad6gacqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaaaaacaGL7baaaaa@4600@  , 

                   F(x)= 1 2 e x [ xsinx+( −x+1 )cosx ] MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaacIcacaWG4bGaaiykaiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaGaamyzamaaCaaaleqabaGaamiEaaaakmaadmaabaGaamiEaiGacohacaGGPbGaaiOBaiaadIhacqGHRaWkdaqadaqaaiabgkHiTiaadIhacqGHRaWkcaaIXaaacaGLOaGaayzkaaGaci4yaiaac+gacaGGZbGaamiEaaGaay5waiaaw2faaaaa@4E45@

  1. Fie f:I→R,f(x)= asinx+bcosx msinx+ncosx ,a,b,m,n∈R MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdacaWGjbGaeyOKH4QaamOuaiaacYcacaWGMbGaaiikaiaadIhacaGGPaGaeyypa0ZaaSaaaeaacaWGHbGaci4CaiaacMgacaGGUbGaamiEaiabgUcaRiaadkgaciGGJbGaai4BaiaacohacaWG4baabaGaamyBaiGacohacaGGPbGaaiOBaiaadIhacqGHRaWkcaWGUbGaci4yaiaac+gacaGGZbGaamiEaaaacaGGSaGaamyyaiaacYcacaWGIbGaaiilaiaad2gacaGGSaGaamOBaiabgIGiolaadkfaaaa@5DC2@ , iar I un interval de numere reale astfel incat asinx+bcosx≠0,( ∀ )x∈I MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiGacohacaGGPbGaaiOBaiaadIhacqGHRaWkcaWGIbGaci4yaiaac+gacaGGZbGaamiEaiabgcMi5kaaicdacaGGSaWaaeWaaeaacqGHaiIiaiaawIcacaGLPaaacaWG4bGaeyicI4Saamysaaaa@4920@ . Exista α,β∈R MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaaiilaiabek7aIjabgIGiolaadkfaaaa@3C3E@  astfel incat f(x)=α+β u'(x) u(x) MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaiabg2da9iabeg7aHjabgUcaRiabek7aInaalaaabaGaamyDaiaacEcacaGGOaGaamiEaiaacMcaaeaacaWG1bGaaiikaiaadIhacaGGPaaaaaaa@45B7@ , unde

    u(x)=msinx+ncosx MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaiaacIcacaWG4bGaaiykaiabg2da9iaad2gaciGGZbGaaiyAaiaac6gacaWG4bGaey4kaSIaamOBaiGacogacaGGVbGaai4CaiaadIhaaaa@44B5@ . Atunci, functia F:I→R,F(x)=αx+βln| u(x) | MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaacQdacaWGjbGaeyOKH4QaamOuaiaacYcacaWGgbGaaiikaiaadIhacaGGPaGaeyypa0JaeqySdeMaamiEaiabgUcaRiabek7aIjGacYgacaGGUbWaaqWaaeaacaWG1bGaaiikaiaadIhacaGGPaaacaGLhWUaayjcSdaaaa@4D5A@  este o primitiva a

    functiei f.

Exemplu: Fie f:I→R,f(x)= 3sinx−cosx sinx−cosx ,I=( − π 4 , π 4 ) MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdacaWGjbGaeyOKH4QaamOuaiaacYcacaWGMbGaaiikaiaadIhacaGGPaGaeyypa0ZaaSaaaeaacaaIZaGaci4CaiaacMgacaGGUbGaamiEaiabgkHiTiGacogacaGGVbGaai4CaiaadIhaaeaaciGGZbGaaiyAaiaac6gacaWG4bGaeyOeI0Iaci4yaiaac+gacaGGZbGaamiEaaaacaGGSaGaamysaiabg2da9maabmaabaGaeyOeI0YaaSaaaeaacqaHapaCaeaacaaI0aaaaiaacYcadaWcaaqaaiabec8aWbqaaiaaisdaaaaacaGLOaGaayzkaaaaaa@5CD6@ .

Obtinem: f(x)=2+ u'(x) u(x) MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaiabg2da9iaaikdacqGHRaWkdaWcaaqaaiaadwhacaGGNaGaaiikaiaadIhacaGGPaaabaGaamyDaiaacIcacaWG4bGaaiykaaaaaaa@4333@ , u(x)=sinx−cosx MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaiaacIcacaWG4bGaaiykaiabg2da9iGacohacaGGPbGaaiOBaiaadIhacqGHsislciGGJbGaai4BaiaacohacaWG4baaaa@42DB@  si F:I→R,F(x)=2x+ln| sinx−cosx | MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaacQdacaWGjbGaeyOKH4QaamOuaiaacYcacaWGgbGaaiikaiaadIhacaGGPaGaeyypa0JaaGOmaiaadIhacqGHRaWkciGGSbGaaiOBamaaemaabaGaci4CaiaacMgacaGGUbGaamiEaiabgkHiTiGacogacaGGVbGaai4CaiaadIhaaiaawEa7caGLiWoaaaa@5018@

Observatii:

§  Exemplele rezolvate fac parte din exercitiile propuse in manualele alternative de Analiza matematica pentru clasa a XII-a, capitolul „ MathType@MTEF@5@5@+=feaagyart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqaaaaaaaaaWdbiaa=5biaaa@37CB@ Integrarea prin parti”, sau au fost propuse la concursuri scolare si in Gazeta matematica,

§  Cititorul poate sa aplice metoda expusa si la alte exemple si poate obtine si generalizari, ca de exemplu pentru f:( 0,∞ )→R,f(x)= x α ln k x,α∈R−{ −1 },k∈ N ∗ MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdadaqadaqaaiaaicdacaGGSaGaeyOhIukacaGLOaGaayzkaaGaeyOKH4QaamOuaiaacYcacaWGMbGaaiikaiaadIhacaGGPaGaeyypa0JaamiEamaaCaaaleqabaGaeqySdegaaOGaciiBaiaac6gadaahaaWcbeqaaiaadUgaaaGccaWG4bGaaiilaiabeg7aHjabgIGiolaadkfacqGHsisldaGadaqaaiabgkHiTiaaigdaaiaawUhacaGL9baacaGGSaGaam4AaiabgIGiolaad6eadaahaaWcbeqaaiabgEHiQaaaaaa@5919@  sau f:R→R,f(x)=( asinαx+bcosαx ) e βx ,a,b,α,β∈R MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdacaWGsbGaeyOKH4QaamOuaiaacYcacaWGMbGaaiikaiaadIhacaGGPaGaeyypa0ZaaeWaaeaacaWGHbGaci4CaiaacMgacaGGUbGaeqySdeMaamiEaiabgUcaRiaadkgaciGGJbGaai4BaiaacohacqaHXoqycaWG4baacaGLOaGaayzkaaGaamyzamaaCaaaleqabaGaeqOSdiMaamiEaaaakiaacYcacaWGHbGaaiilaiaadkgacaGGSaGaeqySdeMaaiilaiabek7aIjabgIGiolaadkfaaaa@5D30@

§  Se pot elabora programe pe calculator pentru rezolvarea unor probleme de calcul al primitivelor, ca de exemplu:

1.b. Generalizare:

.Fie f:R→R,f(x)=( x n + a 1 x n−1 +...+ a n ) e x MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdacaWGsbGaeyOKH4QaamOuaiaacYcacaWGMbGaaiikaiaadIhacaGGPaGaeyypa0JaaiikaiaadIhadaahaaWcbeqaaiaad6gaaaGccqGHRaWkcaWGHbWaaSbaaSqaaiaaigdaaeqaaOGaamiEamaaCaaaleqabaGaamOBaiabgkHiTiaaigdaaaGccqGHRaWkcaGGUaGaaiOlaiaac6cacqGHRaWkcaWGHbWaaSbaaSqaaiaad6gaaeqaaOGaaiykaiaadwgadaahaaWcbeqaaiaadIhaaaaaaa@5233@ , a 1 , a 2 ,... a n ∈R MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBaaaleaacaaIXaaabeaakiaacYcacaWGHbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaac6cacaGGUaGaaiOlaiaadggadaWgaaWcbaGaamOBaaqabaGccqGHiiIZcaWGsbaaaa@4182@ . Sa se determine  b 1 , b 2 ,... b n ∈R MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaaIXaaabeaakiaacYcacaWGIbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaac6cacaGGUaGaaiOlaiaadkgadaWgaaWcbaGaamOBaaqabaGccqGHiiIZcaWGsbaaaa@4185@ , astfel incat functia F:R→R,F(x)=( x n + b 1 x n−1 +...+ b n ) e x MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaacQdacaWGsbGaeyOKH4QaamOuaiaacYcacaWGgbGaaiikaiaadIhacaGGPaGaeyypa0JaaiikaiaadIhadaahaaWcbeqaaiaad6gaaaGccqGHRaWkcaWGIbWaaSbaaSqaaiaaigdaaeqaaOGaamiEamaaCaaaleqabaGaamOBaiabgkHiTiaaigdaaaGccqGHRaWkcaGGUaGaaiOlaiaac6cacqGHRaWkcaWGIbWaaSbaaSqaaiaad6gaaeqaaOGaaiykaiaadwgadaahaaWcbeqaaiaadIhaaaaaaa@51F5@  sa fie o primitiva a functiei f.

2. Fie f:( 0,∞ )→R,f(x)= x n ln k x MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacQdadaqadaqaaiaaicdacaGGSaGaeyOhIukacaGLOaGaayzkaaGaeyOKH4QaamOuaiaacYcacaWGMbGaaiikaiaadIhacaGGPaGaeyypa0JaamiEamaaCaaaleqabaGaamOBaaaakiGacYgacaGGUbWaaWbaaSqabeaacaWGRbaaaOGaamiEaaaa@49EA@ ,  n,k∈ N ∗ MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiaacYcacaWGRbGaeyicI4SaamOtamaaCaaaleqabaGaey4fIOcaaaaa@3BF9@ . Sa se determine  b 1 , b 2 ,... b k ∈R MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaaIXaaabeaakiaacYcacaWGIbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaac6cacaGGUaGaaiOlaiaadkgadaWgaaWcbaGaam4AaaqabaGccqGHiiIZcaWGsbaaaa@4182@ , astfel

               incat functia F:( 0,∞ )→R,F(x)= x n+1 n+1 ( ln k x+ b 1 ln k−1 x+...+ b k ) MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaacQdadaqadaqaaiaaicdacaGGSaGaeyOhIukacaGLOaGaayzkaaGaeyOKH4QaamOuaiaacYcacaWGgbGaaiikaiaadIhacaGGPaGaeyypa0ZaaSaaaeaacaWG4bWaaWbaaSqabeaacaWGUbGaey4kaSIaaGymaaaaaOqaaiaad6gacqGHRaWkcaaIXaaaaiaacIcaciGGSbGaaiOBamaaCaaaleqabaGaam4AaaaakiaadIhacqGHRaWkcaWGIbWaaSbaaSqaaiaaigdaaeqaaOGaciiBaiaac6gadaahaaWcbeqaaiaadUgacqGHsislcaaIXaaaaOGaamiEaiabgUcaRiaac6cacaGGUaGaaiOlaiabgUcaRiaadkgadaWgaaWcbaGaam4AaaqabaGccaGGPaaaaa@5D91@  sa fie o primitiva a

               functiei f.

          Obtinem urmatoarele programe Pascal pentru determinarea coeficientilor :

               b 1 , b 2 ,... b n MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaaIXaaabeaakiaacYcacaWGIbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaac6cacaGGUaGaaiOlaiaadkgadaWgaaWcbaGaamOBaaqabaaaaa@3F20@ , (1.b),                                                     b 1 , b 2 ,... b k MathType@MTEF@5@5@+=feaagyart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBaaaleaacaaIXaaabeaakiaacYcacaWGIbWaaSbaaSqaaiaaikdaaeqaaOGaaiilaiaac6cacaGGUaGaaiOlaiaadkgadaWgaaWcbaGaam4Aaaqabaaaaa@3F1D@ ,   (2.)    

        Program Primitive;                                                   Program Primitive;

uses crt;                                                                uses crt;

var n,i:integer;                                                     var n,k,i:integer;

      a,b:array[0..100] of integer;                            b:array[0..100] of real;

begin                                                                   begin

ClrScr;                                                                  ClrScr;

write(‘n=’);read(n);                                              write(‘n=’);read(n); write(‘k=’);read(k);

for i:=1 to n do                                                    b[0]:=1

     begin                                                              for i:=1 to k do

     write(‘a[‘,i,’]=’);read(a[i]);                                 begin

     end.                                                                     b[i]:=-(k-i+1)*b[i-1]/(n+1);          

b[0]:=0;                                                                    write(‘b[‘,i,]=’,b[i],’  ‘);

for i:=1 to n do                                                         end;

     begin                                                              readln;readln;

     b[i]:=a[i]-(n-i+1)*b[i-1];                                 end.

     write(‘b[‘,i,]=’,b[i],’  ‘);

     end;

readln;readln;

end.

§ Se pot calcula primitive utilizand softul interactiv Maple.

Exemple:

1.      Int(x^2-4*x+3,x)=int(x^2-4*x+3,x);

                                      

2.      Int((x^3*ln(x)^2,x))=collect((int(x^3*ln(x)^2,x),x));

                                     

3.      Int((x^2-x+1)*sin(x),x)=collect(int((x^2-x+1)*sin(x),x),cos(x));

                      

4.      Int(x*ln(x),x)=collect(int(x*ln(x),x),x);

                                    

5.      Int(sin(x)^3*cos(x)^2,x)=int(sin(x)^3*cos(x)^2,x);

Bibliografie:

1.      Analiza matematica pentru clasa a XII-a, Manuale alternative,

2.      Gazeta matematica, Editia electronica,

3.      Analiza matematica cu MAPLE, Lica Dionis, Bucuresti, 2003,

4.      http://mateinfo.ro/