Primitives of Rational Functions (Partial Fraction Decomposition)


This part deals with only one subject: How to find primitives of rational functions, i.e. of functions of the type f= r s MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabg2da9maalaaabaGaamOCaaqaaiaadohaaaaaaa@39DC@ with polynomials r and s. Without restriction we assume s to be non constant and normalized.

On intervals, for example between two adjacent zeros of s, f has a primitive as it is continuous. There is a concept to calculate these primitives which is essentially related to two basic theorems on polynomials:

  • Fundamental Theorem of Algebra

    Each non constant normalized polynomial s on MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@375C@ completely decomposes into a product of linear and indecomposable quadratic polynomials:

    s= (X a 1 ) l 1 (X a j ) l j ( X 2 + p 1 X+ q 1 ) n 1 ( X 2 + p k X+ q k ) n k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6C03@ [1]

    where the discriminant of a quadratic polynomial controls its decomposability:

    X 2 + p i X+ q i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadchadaWgaaWcbaGaamyAaaqabaGccaWGybGaey4kaSIaamyCamaaBaaaleaacaWGPbaabeaaaaa@3E86@ is indecomposable D i = p i 2 4 q i <0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGzbVlabgsDiBlaaywW7caWGebWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0ZaaSaaaeaacaWGWbWaa0baaSqaaiaadMgaaeaacaaIYaaaaaGcbaGaaGinaaaacqGHsislcaWGXbWaaSbaaSqaaiaadMgaaeqaaOGaeyipaWJaaGimaaaa@46C0@

    So we know that X 2 + p i X+ q i (0)= q i >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadchadaWgaaWcbaGaamyAaaqabaGccaWGybGaey4kaSIaamyCamaaBaaaleaacaWGPbaabeaakiaacIcacaaIWaGaaiykaiabg2da9iaadghadaWgaaWcbaGaamyAaaqabaGccqGH+aGpcaaIWaaaaa@4585@ and that, consequently, all values of X 2 + p i X+ q i MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadchadaWgaaWcbaGaamyAaaqabaGccaWGybGaey4kaSIaamyCamaaBaaaleaacaWGPbaabeaaaaa@3E86@ are positive. (Continuous functions with no zeros do not vary their algebraic sign on intervals!)
     

  • Partial Fraction Theorem

    Based on the decomposition [1] each rational function with a non constant mormalized denominator s is representable as a sum:

    r s =t + c 11 X a 1 + c 12 (X a 1 ) 2 ++ c 1 l 1 (X a 1 ) l 1 + + c j1 X a j + c j2 (X a j ) 2 ++ c j l j (X a j ) l j + m 11 X+ b 11 X 2 + p 1 X+ q 1 + m 12 X+ b 12 ( X 2 + p 1 X+ q 1 ) 2 ++ m 1 n 1 X+ b 1 n 1 ( X 2 + p 1 X+ q 1 ) n 1 + + m k1 X+ b k1 X 2 + p k X+ q k + m k2 X+ b k2 ( X 2 + p k X+ q k ) 2 ++ m k n k X+ b k n k ( X 2 + p k X+ q k ) n k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@F454@

    t being a suitable polynomial such that t=0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDaiabg2da9iaaicdaaaa@38A5@ if grad   r<grad   s MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4zaiaabkhacaqGHbGaaeizaiaaysW7caWGYbGaeyipaWJaae4zaiaabkhacaqGHbGaaeizaiaaysW7caWGZbaaaa@434D@ and grad   t=grad   rgrad   s MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4zaiaabkhacaqGHbGaaeizaiaaysW7caWG0bGaeyypa0Jaae4zaiaabkhacaqGHbGaaeizaiaaysW7caWGYbGaeyOeI0Iaae4zaiaabkhacaqGHbGaaeizaiaaysW7caWGZbaaaa@4A6C@ otherweise.
     

Both theorems, especially the fundamental theorem, are pure existence theorems! And it is this fact that causes actual problems when it comes to application: Apart from simple and clear cut cases it is nearly impossible to get the necessary denominator's decomposition.

Example:  We find a partial decomposition for

f= 2 X 5 4 X 4 +10 X 3 17 X 2 +6X3 X 4 2 X 3 +3 X 2 4X+2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabg2da9maalaaabaGaaGOmaiaadIfadaahaaWcbeqaaiaaiwdaaaGccqGHsislcaaI0aGaamiwamaaCaaaleqabaGaaGinaaaakiabgUcaRiaaigdacaaIWaGaamiwamaaCaaaleqabaGaaG4maaaakiabgkHiTiaaigdacaaI3aGaamiwamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaiAdacaWGybGaeyOeI0IaaG4maaqaaiaadIfadaahaaWcbeqaaiaaisdaaaGccqGHsislcaaIYaGaamiwamaaCaaaleqabaGaaG4maaaakiabgUcaRiaaiodacaWGybWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGinaiaadIfacqGHRaWkcaaIYaaaaaaa@5764@
 
  1. First we (sequentially) discover that 1 is a double zero of the denominator so that a twofold polynomial division yields the decomposition

    X 4 2 X 3 +3 X 2 4X+2= (X1) 2 ( X 2 +2) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaCaaaleqabaGaaGinaaaakiabgkHiTiaaikdacaWGybWaaWbaaSqabeaacaaIZaaaaOGaey4kaSIaaG4maiaadIfadaahaaWcbeqaaiaaikdaaaGccqGHsislcaaI0aGaamiwaiabgUcaRiaaikdacqGH9aqpcaGGOaGaamiwaiabgkHiTiaaigdacaGGPaWaaWbaaSqabeaacaaIYaaaaOGaeyyXICTaaiikaiaadIfadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIYaGaaiykaaaa@4FB5@
     
  2. and a further polynomial division then provides the identity

    f=2X+ 4 X 3 9 X 2 +2X3 (X1) 2 ( X 2 +2) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabg2da9iaaikdacaWGybGaey4kaSYaaSaaaeaacaaI0aGaamiwamaaCaaaleqabaGaaG4maaaakiabgkHiTiaaiMdacaWGybWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGOmaiaadIfacqGHsislcaaIZaaabaGaaiikaiaadIfacqGHsislcaaIXaGaaiykamaaCaaaleqabaGaaGOmaaaakiabgwSixlaacIcacaWGybWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGOmaiaacMcaaaaaaa@507E@
  3. Finally, comparing the coefficients in the ansatz

    4 X 3 9 X 2 +2X3 (X1) 2 ( X 2 +2) = c 11 X1 + c 12 (X1) 2 + m 11 X+ b 11 X 2 +2 = c 11 (X1)( X 2 +2)+ c 12 ( X 2 +2)+( m 11 X+ b 11 ) (X1) 2 (X1) 2 ( X 2 +2) = ( c 11 + m 11 ) X 3 +( c 11 + c 12 2 m 11 + b 11 ) X 2 +(2 c 11 + m 11 2 b 11 )X2 c 11 +2 c 12 + b 11 (X1) 2 ( X 2 +2) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@D2D3@

    yields the following linear system of equations

    c 11 + m 11 =4 c 11 =0 c 11 + c 12 2 m 11 + b 11 =9 c 12 =2 2 c 11 + m 11 2 b 11 =2 m 11 =4 2 c 11 +2 c 12 + b 11 =3 b 11 =1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabqqbaaaaaeaacaWGJbWaaSbaaSqaaiaaigdacaaIXaaabeaakiabgUcaRiaad2gadaWgaaWcbaGaaGymaiaaigdaaeqaaaGcbaGaeyypa0JaaGinaaqaaiaaywW7cqGHuhY2caaMf8oabaGaam4yamaaBaaaleaacaaIXaGaaGymaaqabaaakeaacqGH9aqpcaaIWaaabaGaeyOeI0Iaam4yamaaBaaaleaacaaIXaGaaGymaaqabaGccqGHRaWkcaWGJbWaaSbaaSqaaiaaigdacaaIYaaabeaakiabgkHiTiaaikdacaWGTbWaaSbaaSqaaiaaigdacaaIXaaabeaakiabgUcaRiaadkgadaWgaaWcbaGaaGymaiaaigdaaeqaaaGcbaGaeyypa0JaeyOeI0IaaGyoaaqaaaqaaiaadogadaWgaaWcbaGaaGymaiaaikdaaeqaaaGcbaGaeyypa0JaeyOeI0IaaGOmaaqaaiaaikdacaWGJbWaaSbaaSqaaiaaigdacaaIXaaabeaakiabgUcaRiaad2gadaWgaaWcbaGaaGymaiaaigdaaeqaaOGaeyOeI0IaaGOmaiaadkgadaWgaaWcbaGaaGymaiaaigdaaeqaaaGcbaGaeyypa0JaaGOmaaqaaaqaaiaad2gadaWgaaWcbaGaaGymaiaaigdaaeqaaaGcbaGaeyypa0JaaGinaaqaaiabgkHiTiaaikdacaWGJbWaaSbaaSqaaiaaigdacaaIXaaabeaakiabgUcaRiaaikdacaWGJbWaaSbaaSqaaiaaigdacaaIYaaabeaakiabgUcaRiaadkgadaWgaaWcbaGaaGymaiaaigdaaeqaaaGcbaGaeyypa0JaeyOeI0IaaG4maaqaaaqaaiaadkgadaWgaaWcbaGaaGymaiaaigdaaeqaaaGcbaGaeyypa0JaaGymaaaaaaa@827B@

    f thus decomposes like this

    f=2X 2 (X1) 2 + 4X+1 X 2 +2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabg2da9iaaikdacaWGybGaeyOeI0YaaSaaaeaacaaIYaaabaGaaiikaiaadIfacqGHsislcaaIXaGaaiykamaaCaaaleqabaGaaGOmaaaaaaGccqGHRaWkdaWcaaqaaiaaisdacaWGybGaey4kaSIaaGymaaqaaiaadIfadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIYaaaaaaa@4798@ [2]

With a partial decomposition of f at hand, a primitive of f is easily found if we know primitives for each addend. This reduces our problem to only two types of rational functions, namely ( k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgIGiolablwriLcaa@39CC@ )

  1. c (Xa) k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGJbaabaGaaiikaiaadIfacqGHsislcaWGHbGaaiykamaaCaaaleqabaGaam4Aaaaaaaaaaa@3C0A@

  2. mX+b ( X 2 +pX+q ) k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGTbGaamiwaiabgUcaRiaadkgaaeaacaGGOaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadchacaWGybGaey4kaSIaamyCaiaacMcadaahaaWcbeqaaiaadUgaaaaaaaaa@4266@   where  D= p 2 4 q<0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabg2da9maalaaabaGaamiCamaaCaaaleqabaGaaGOmaaaaaOqaaiaaisdaaaGaeyOeI0IaamyCaiabgYda8iaaicdaaaa@3E12@

Quotients of the first kind are easy to deal with, except for the case k=1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg2da9iaaigdaaaa@389D@ which needs the natural logarithm ln

 i

from chapter 8.

Proposition:  

  1. cln|Xa| MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiabgwSixlGacYgacaGGUbGaaiiFaiaadIfacqGHsislcaWGHbGaaiiFaaaa@3FB2@   is a primitive function of  c Xa MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGJbaabaGaamiwaiabgkHiTiaadggaaaaaaa@3994@ .

[8.0.1]
  1. If k>1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg6da+iaaigdaaaa@389F@ then  1 1k c (Xa) k1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaaGymaiabgkHiTiaadUgaaaGaeyyXIC9aaSaaaeaacaWGJbaabaGaaiikaiaadIfacqGHsislcaWGHbGaaiykamaaCaaaleqabaGaam4AaiabgkHiTiaaigdaaaaaaaaa@435F@   is a primitive function of  c (Xa) k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGJbaabaGaaiikaiaadIfacqGHsislcaWGHbGaaiykamaaCaaaleqabaGaam4Aaaaaaaaaaa@3C0A@ .

[8.0.2]

Proof:  

1. ►  In [8.7.1] we introduce ln as a primitive of the reciprocal function on >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaWbaaSqabeaacqGH+aGpcaaIWaaaaaaa@394B@ . ln is thus differentiable and

ln(x)= 1 x MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaGGNaGaaiikaiaadIhacaGGPaGaeyypa0ZaaSaaaeaacaaIXaaabaGaamiEaaaaaaa@3D9F@   for all x>0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabg6da+iaaicdaaaa@38AB@

Using the chain rule [7.7.8] and the derivative of the absolute value function [7.4.3] we see that cln|Xa| MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiabgwSixlGacYgacaGGUbGaaiiFaiaadIfacqGHsislcaWGHbGaaiiFaaaa@3FB2@ is differentiable and that

(cln|Xa| ) =c 1 |Xa| |Xa | ) =c 1 |Xa| |Xa| Xa = c Xa MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadogacqGHflY1ciGGSbGaaiOBaiaacYhacaWGybGaeyOeI0IaamyyaiaacYhaceGGPaGbauaacqGH9aqpcaWGJbGaeyyXIC9aaSaaaeaacaaIXaaabaGaaiiFaiaadIfacqGHsislcaWGHbGaaiiFaaaacqGHflY1caGG8bGaamiwaiabgkHiTiaadggaceGG8bGbauaacqGH9aqpcaWGJbGaeyyXIC9aaSaaaeaacaaIXaaabaGaaiiFaiaadIfacqGHsislcaWGHbGaaiiFaaaacqGHflY1daWcaaqaaiaacYhacaWGybGaeyOeI0IaamyyaiaacYhaaeaacaWGybGaeyOeI0IaamyyaaaacqGH9aqpdaWcaaqaaiaadogaaeaacaWGybGaeyOeI0Iaamyyaaaaaaa@69EB@

is its derivative.

2. ►   1 1k c (Xa) k1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaaGymaiabgkHiTiaadUgaaaGaeyyXIC9aaSaaaeaacaWGJbaabaGaaiikaiaadIfacqGHsislcaWGHbGaaiykamaaCaaaleqabaGaam4AaiabgkHiTiaaigdaaaaaaaaa@435F@ is essentially a power of X, thus differentiable. We use the power rule

 i

( X n ) =n X n1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIfadaahaaWcbeqaaiaad6gaaaGcceGGPaGbauaacqGH9aqpcaWGUbGaamiwamaaCaaaleqabaGaamOBaiabgkHiTiaaigdaaaaaaa@3EF6@ for all n MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaiabgIGiolablssiIcaa@39DB@

to calculate the derivative:

( 1 1k c (Xa) k1 ) = 1 1k ((k1)) c (Xa) k = c (Xa) k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@614E@

Examples are quite straight usually:

  • 4ln|X7| MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiabgwSixlGacYgacaGGUbGaaiiFaiaadIfacqGHsislcaaI3aGaaiiFaaaa@3F63@   is a primitive of  4 X7 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaI0aaabaGaamiwaiabgkHiTiaaiEdaaaaaaa@3945@ .

  • 2 X1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaSaaaeaacaaIYaaabaGaamiwaiabgkHiTiaaigdaaaaaaa@3A2A@   is a primitive of  2 (X1) 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIYaaabaGaaiikaiaadIfacqGHsislcaaIXaGaaiykamaaCaaaleqabaGaaGOmaaaaaaaaaa@3B7F@ .
     

Dealing with quotients of the second kind is a bit catchier. But as we always have the decomposition

mX+b ( X 2 +pX+q ) k = m 2 2X+p ( X 2 +pX+q ) k + b m 2 p ( X 2 +pX+q ) k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@6225@ [3]

we may confine ourselves only to the cases 2X+p ( X 2 +pX+q ) k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIYaGaamiwaiabgUcaRiaadchaaeaacaGGOaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadchacaWGybGaey4kaSIaamyCaiaacMcadaahaaWcbeqaaiaadUgaaaaaaaaa@423E@ and c ( X 2 +pX+q ) k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGJbaabaGaaiikaiaadIfadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGWbGaamiwaiabgUcaRiaadghacaGGPaWaaWbaaSqabeaacaWGRbaaaaaaaaa@3FB6@ . The first one is easy.

Proposition:  

  1. ln( X 2 +pX+q) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaGGOaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadchacaWGybGaey4kaSIaamyCaiaacMcaaaa@3F85@   is a primitive function of  2X+p X 2 +pX+q MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIYaGaamiwaiabgUcaRiaadchaaeaacaWGybWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamiCaiaadIfacqGHRaWkcaWGXbaaaaaa@3FC8@ .

[8.0.3]
  1. If k>1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg6da+iaaigdaaaa@389F@ then

    1 1k 1 ( X 2 +pX+q ) k1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaaGymaiabgkHiTiaadUgaaaGaeyyXIC9aaSaaaeaacaaIXaaabaGaaiikaiaadIfadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGWbGaamiwaiabgUcaRiaadghacaGGPaWaaWbaaSqabeaacaWGRbGaeyOeI0IaaGymaaaaaaaaaa@46DE@   is a primitive function of  2X+p ( X 2 +pX+q ) k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIYaGaamiwaiabgUcaRiaadchaaeaacaGGOaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadchacaWGybGaey4kaSIaamyCaiaacMcadaahaaWcbeqaaiaadUgaaaaaaaaa@423E@ .

[8.0.4]

Proof:  Both assertions are easily proved by using the chain rule. In 1. we note that, due to the premise, the values of X 2 +pX+q MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadchacaWGybGaey4kaSIaamyCaaaa@3C48@ are always >0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOpa4JaaGimaaaa@37AE@ . The domain of ln( X 2 +pX+q) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaGGOaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaadchacaWGybGaey4kaSIaamyCaiaacMcaaaa@3F85@ is thus the whole of MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa@375C@ .

As an example we see that

  • ln( X 2 +2) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6gacaGGOaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaikdacaGGPaaaaa@3C97@   is a primitive function of  2X X 2 +2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIYaGaamiwaaqaaiaadIfadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIYaaaaaaa@3B03@ .

  • 1 3 1 ( X 2 3X+5 ) 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaSaaaeaacaaIXaaabaGaaG4maaaacqGHflY1daWcaaqaaiaaigdaaeaacaGGOaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaiodacaWGybGaey4kaSIaaGynaiaacMcadaahaaWcbeqaaiaaiodaaaaaaaaa@43B1@   is a primitive function of  2X3 ( X 2 3X+5 ) 4 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIYaGaamiwaiabgkHiTiaaiodaaeaacaGGOaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaiodacaWGybGaey4kaSIaaGynaiaacMcadaahaaWcbeqaaiaaisdaaaaaaaaa@417B@ .
     

The remaining case is actually quite cumbersome. We recall that the discriminant D= p 2 4 q MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabg2da9maalaaabaGaamiCamaaCaaaleqabaGaaGOmaaaaaOqaaiaaisdaaaGaeyOeI0IaamyCaaaa@3C54@ is negative, thus D MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaamiraaaa@37A2@ is positive. At first we show that it is sufficient to consider only denominators like ( X 2 +1 ) k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIfadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIXaGaaiykamaaCaaaleqabaGaam4Aaaaaaaa@3BCF@ .

Proposition:  If g is a primitive of c ( X 2 +1 ) k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGJbaabaGaaiikaiaadIfadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIXaGaaiykamaaCaaaleqabaGaam4Aaaaaaaaaaa@3CC7@ then

D 12k g X+ p 2 D MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacqGHsislcaWGebWaaWbaaSqabeaacaaIXaGaeyOeI0IaaGOmaiaadUgaaaaabeaakiabgwSixlaadEgacqWIyiYBdaWcaaqaaiaadIfacqGHRaWkdaWcaaqaaiaadchaaeaacaaIYaaaaaqaamaakaaabaGaeyOeI0IaamiraaWcbeaaaaaaaa@450E@
[8.0.5]

is a primitive function of c ( X 2 +pX+q ) k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGJbaabaGaaiikaiaadIfadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGWbGaamiwaiabgUcaRiaadghacaGGPaWaaWbaaSqabeaacaWGRbaaaaaaaaa@3FB6@ .

Proof:  The chain rule guarantees the differentiability of the function in [8.0.5] and provides the following derivative:

( D 12k g X+ p 2 D ) = (D) 12k ( g X+ p 2 D ) 1 D = (D) 2k c ( X 2 +1 ) k X+ p 2 D = (D) k c ( (X+ p 2 ) 2 D +1) k = (D) k c (D) k ( X 2 +pX+ p 2 4 D) k = c ( X 2 +pX+q ) k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@B165@

Primitives of ( X 2 +1 ) k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadIfadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIXaGaaiykamaaCaaaleqabaGaam4Aaaaaaaa@3BCF@ are eventually constructed by recursion. For the base step we need the invers tangent,

arctan:] π 2 , π 2 [ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyyaiaackhacaGGJbGaaiiDaiaacggacaGGUbGaaiOoaiabl2riHkabgkziUkaac2facqGHsisldaWcaaqaaiabec8aWbqaaiaaikdaaaGaaiilamaalaaabaGaeqiWdahabaGaaGOmaaaacaGGBbaaaa@4809@ ,

the inverse of tan|] π 2 , π 2 [ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiDaiaacggacaGGUbGaaiiFaiaac2facqGHsisldaWcaaqaaiabec8aWbqaaiaaikdaaaGaaiilamaalaaabaGaeqiWdahabaGaaGOmaaaacaGGBbaaaa@422C@

 i

The restriction tan|] π 2 , π 2 [ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiDaiaacggacaGGUbGaaiiFaiaac2facqGHsisldaWcaaqaaiabec8aWbqaaiaaikdaaaGaaiilamaalaaabaGaeqiWdahabaGaaGOmaaaacaGGBbaaaa@422C@ is bijective as it is

  • injective due to [7.9.6] because

    tan(x)= 1 cos 2 (x) 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiDaiaacggacaGGUbGaai4jaiaacIcacaWG4bGaaiykaiabg2da9maalaaabaGaaGymaaqaaiGacogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaakiaacIcacaWG4bGaaiykaaaacqGHGjsUcaaIWaaaaa@462C@ for all x] π 2 , π 2 [ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaac2facqGHsisldaWcaaqaaiabec8aWbqaaiaaikdaaaGaaiilamaalaaabaGaeqiWdahabaGaaGOmaaaacaGGBbaaaa@40DC@ .
     
  • surjective according to a consequence of the intermediate value theorem [6.6.2] based on the limits

    lim x± π 2 x] π 2 , π 2 [ tanx=± MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaciGGSbGaaiyAaiaac2gaaSabaeqabaGaamiEaiabgkziUkabgglaXoaalaaabaGaeqiWdahabaGaaGOmaaaaaeaacaWG4bGaeyicI4SaaiyxaiabgkHiTmaalaaabaGaeqiWdahabaGaaGOmaaaacaGGSaWaaSaaaeaacqaHapaCaeaacaaIYaaaaiaacUfaaaqabaGcciGG0bGaaiyyaiaac6gacaWG4bGaeyypa0JaeyySaeRaeyOhIukaaa@538A@
.

Proposition:  

  1. carctan MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaiabgwSixlGacggacaGGYbGaai4yaiaacshacaGGHbGaaiOBaaaa@3EB1@   is a primitive function of  c X 2 +1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGJbaabaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaigdaaaaaaa@3A51@ .

[8.0.6]
  1. If g is a primitive of  c ( X 2 +1 ) k MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGJbaabaGaaiikaiaadIfadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIXaGaaiykamaaCaaaleqabaGaam4Aaaaaaaaaaa@3CC7@ then

1 2k ( cX ( X 2 +1 ) k +(2k1)g) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIXaaabaGaaGOmaiaadUgaaaGaeyyXICTaaiikamaalaaabaGaam4yaiaadIfaaeaacaGGOaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaigdacaGGPaWaaWbaaSqabeaacaWGRbaaaaaakiabgUcaRiaacIcacaaIYaGaam4AaiabgkHiTiaaigdacaGGPaGaeyyXICTaam4zaiaacMcaaaa@4C8D@
[8.0.7]

is a primitive function of  c ( X 2 +1 ) k+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGJbaabaGaaiikaiaadIfadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIXaGaaiykamaaCaaaleqabaGaam4AaiabgUcaRiaaigdaaaaaaaaa@3E64@ .

Proof:  

1. ►  As tan(x)= 1 cos 2 (x) 0 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiDaiaacggacaGGUbGaai4jaiaacIcacaWG4bGaaiykaiabg2da9maalaaabaGaaGymaaqaaiGacogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaakiaacIcacaWG4bGaaiykaaaacqGHGjsUcaaIWaaaaa@462C@ for all x] π 2 , π 2 [ MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgIGiolaac2facqGHsisldaWcaaqaaiabec8aWbqaaiaaikdaaaGaaiilamaalaaabaGaeqiWdahabaGaaGOmaaaacaGGBbaaaa@40DC@ , arctan is differentiable due to [7.5.4] with

arctan(x)= 1 tan(arctanx) = cos 2 (arctanx) = [+] 1 tan 2 (arctanx)+1 = 1 x 2 +1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyyaiaackhacaGGJbGaaiiDaiaacggacaGGUbGaai4jaiaacIcacaWG4bGaaiykaiabg2da9maalaaabaGaaGymaaqaaiGacshacaGGHbGaaiOBaiaacEcacaGGOaGaciyyaiaackhacaGGJbGaaiiDaiaacggacaGGUbGaamiEaiaacMcaaaGaeyypa0Jaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaaIYaaaaOGaaiikaiGacggacaGGYbGaai4yaiaacshacaGGHbGaaiOBaiaadIhacaGGPaWaaCbeaeaacqGH9aqpaSqaaiaacUfacqGHRaWkcaGGDbaabeaakmaalaaabaGaaGymaaqaaiGacshacaGGHbGaaiOBamaaCaaaleqabaGaaGOmaaaakiaacIcaciGGHbGaaiOCaiaacogacaGG0bGaaiyyaiaac6gacaWG4bGaaiykaiabgUcaRiaaigdaaaGaeyypa0ZaaSaaaeaacaaIXaaabaGaamiEamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaigdaaaaaaa@6FC9@

Note that [+] is valid according to the identity cos 2 = 1 tan 2 +1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaaIYaaaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaciiDaiaacggacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGymaaaaaaa@40E4@ .

2. ►  The function in [8.0.7] is differentiable due to the quotient rule [7.7.7]. Its derivative calculates to:

1 2k ( cX ( X 2 +1 ) k +(2k1)g ) = 1 2k ( c ( X 2 +1 ) k cXk ( X 2 +1 ) k1 2X ( X 2 +1 ) 2k +(2k1) c ( X 2 +1 ) k = c 2k X 2 +12k X 2 +(2k1)( X 2 +1) ( X 2 +1 ) k+1 = c 2k 2k ( X 2 +1 ) k+1 = c ( X 2 +1 ) k+1 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@BB8C@

An example will show how this procedure works.

Example:  

  • It takes three steps to construct a primitive function of 2 ( X 2 +1 ) 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIYaaabaGaaiikaiaadIfadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIXaGaaiykamaaCaaaleqabaGaaG4maaaaaaaaaa@3C68@ recursively by [8.0.7]:

    1.)2arctan MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeymaiaab6cacaqGPaGaaGzbVlaaikdacqGHflY1ciGGHbGaaiOCaiaacogacaGG0bGaaiyyaiaac6gaaaa@4224@ is a primitive of 2 ( X 2 +1) MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIYaaabaGaaiikaiaadIfadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIXaGaaiykaaaaaaa@3B7E@ .

    2.) k=1 1 2 ( 2X X 2 +1 +2arctan)= X X 2 +1 +arctan MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaacaqGYaGaaeOlaiaabMcaaSqaaiaadUgacqGH9aqpcaaIXaaabeaakiaaywW7daWcaaqaaiaaigdaaeaacaaIYaaaaiabgwSixlaacIcadaWcaaqaaiaaikdacaWGybaabaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaigdaaaGaey4kaSIaaGOmaiabgwSixlGacggacaGGYbGaai4yaiaacshacaGGHbGaaiOBaiaacMcacqGH9aqpdaWcaaqaaiaadIfaaeaacaWGybWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGymaaaacqGHRaWkciGGHbGaaiOCaiaacogacaGG0bGaaiyyaiaac6gaaaa@5C10@ is a primitive of 2 ( X 2 +1 ) 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIYaaabaGaaiikaiaadIfadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIXaGaaiykamaaCaaaleqabaGaaGOmaaaaaaaaaa@3C67@ .

    3.) k=2 1 4 ( 2X ( X 2 +1 ) 2 +3( X X 2 +1 +arctan))= 1 2 X ( X 2 +1 ) 2 + 3 4 X X 2 +1 + 3 4 arctan MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@77F9@
    is a primitive of 2 ( X 2 +1 ) 3 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIYaaabaGaaiikaiaadIfadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIXaGaaiykamaaCaaaleqabaGaaG4maaaaaaaaaa@3C68@ .
     

  • To get a primitive of 2 ( X 2 +6X+13 ) 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIYaaabaGaaiikaiaadIfadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaI2aGaamiwaiabgUcaRiaaigdacaaIZaGaaiykamaaCaaaleqabaGaaGOmaaaaaaaaaa@3FA3@ we start by calculating the discriminant D=4 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabg2da9iabgkHiTiaaisdaaaa@3966@ from the data p=6 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2da9iaaiAdaaaa@38A7@ and q=13 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyCaiabg2da9iaaigdacaaIZaaaaa@3960@ . With step two of the example above and with [8.0.5] we then find

    D 12k g X+ p 2 D = 1 64 ( X X 2 +1 +arctan) X+3 2 = 1 8 ( X+3 2 (X+3) 2 +4 4 +arctan X+3 2 ) = 1 4 X+3 X 2 +6X+13 + 1 8 arctan X+3 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabmGaaaqaamaakaaabaGaeyOeI0IaamiramaaCaaaleqabaGaaGymaiabgkHiTiaaikdacaWGRbaaaaqabaGccqGHflY1caWGNbGaeSigI82aaSaaaeaacaWGybGaey4kaSYaaSaaaeaacaWGWbaabaGaaGOmaaaaaeaadaGcaaqaaiabgkHiTiaadseaaSqabaaaaaGcbaGaeyypa0ZaaSaaaeaacaaIXaaabaWaaOaaaeaacaaI2aGaaGinaaWcbeaaaaGccqGHflY1caGGOaWaaSaaaeaacaWGybaabaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaigdaaaGaey4kaSIaciyyaiaackhacaGGJbGaaiiDaiaacggacaGGUbGaaiykaiablIHiVnaalaaabaGaamiwaiabgUcaRiaaiodaaeaacaaIYaaaaaqaaaqaaiabg2da9maalaaabaGaaGymaaqaaiaaiIdaaaGaeyyXICTaaiikamaalaaabaWaaSaaaeaacaWGybGaey4kaSIaaG4maaqaaiaaikdaaaaabaWaaSaaaeaacaGGOaGaamiwaiabgUcaRiaaiodacaGGPaWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGinaaqaaiaaisdaaaaaaiabgUcaRiGacggacaGGYbGaai4yaiaacshacaGGHbGaaiOBamaalaaabaGaamiwaiabgUcaRiaaiodaaeaacaaIYaaaaiaacMcaaeaaaeaacqGH9aqpdaWcaaqaaiaaigdaaeaacaaI0aaaaiabgwSixpaalaaabaGaamiwaiabgUcaRiaaiodaaeaacaWGybWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGOnaiaadIfacqGHRaWkcaaIXaGaaG4maaaacqGHRaWkdaWcaaqaaiaaigdaaeaacaaI4aaaaiabgwSixlGacggacaGGYbGaai4yaiaacshacaGGHbGaaiOBamaalaaabaGaamiwaiabgUcaRiaaiodaaeaacaaIYaaaaaaaaaa@91B5@

    as a primitive function.
     

  • Finally, to get a primitive of 3X+11 ( X 2 +6X+13 ) 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaaIZaGaamiwaiabgUcaRiaaigdacaaIXaaabaGaaiikaiaadIfadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaI2aGaamiwaiabgUcaRiaaigdacaaIZaGaaiykamaaCaaaleqabaGaaGOmaaaaaaaaaa@42D9@ we first consider the decomposition

    3X+11 ( X 2 +6X+13 ) 2 = 3 2 2X+6 ( X 2 +6X+13 ) 2 + 2 ( X 2 +6X+13 ) 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5E95@

    according to [3]. From the previous result and from [8.0.4] we now find

    3 2 1 X 2 +6X+13 + 1 4 X+3 X 2 +6X+13 + 1 8 arctan X+3 2 = 1 4 X3 X 2 +6X+13 + 1 8 arctan X+3 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@795A@

    as a primitive.

We now return to our initial example

f= 2 X 5 4 X 4 +10 X 3 17 X 2 +6X3 X 4 2 X 3 +3 X 2 4X+2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiabg2da9maalaaabaGaaGOmaiaadIfadaahaaWcbeqaaiaaiwdaaaGccqGHsislcaaI0aGaamiwamaaCaaaleqabaGaaGinaaaakiabgUcaRiaaigdacaaIWaGaamiwamaaCaaaleqabaGaaG4maaaakiabgkHiTiaaigdacaaI3aGaamiwamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaiAdacaWGybGaeyOeI0IaaG4maaqaaiaadIfadaahaaWcbeqaaiaaisdaaaGccqGHsislcaaIYaGaamiwamaaCaaaleqabaGaaG4maaaakiabgUcaRiaaiodacaWGybWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaaGinaiaadIfacqGHRaWkcaaIYaaaaaaa@5764@

In [2] we showed that

2X 2 (X1) 2 + 4X+1 X 2 +2 =2X 2 (X1) 2 +2 2X X 2 +2 + 1 X 2 +2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5CEA@

is its partial fraction decomposition. Thus f is completely decomposed in processable basic typs and a primitive of f is simply gained by adding their primitive functions:

X 2 + 2 X1 +2ln( X 2 +2)+ 1 2 arctan X 2 MathType@MTEF@5@5@+=feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiwamaaCaaaleqabaGaaGOmaaaakiabgUcaRmaalaaabaGaaGOmaaqaaiaadIfacqGHsislcaaIXaaaaiabgUcaRiaaikdacqGHflY1ciGGSbGaaiOBaiaacIcacaWGybWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGOmaiaacMcacqGHRaWkdaWcaaqaaiaaigdaaeaadaGcaaqaaiaaikdaaSqabaaaaOGaeyyXICTaciyyaiaackhacaGGJbGaaiiDaiaacggacaGGUbWaaSaaaeaacaWGybaabaWaaOaaaeaacaaIYaaaleqaaaaaaaa@52B1@