Expansion of the integral x. integral from the value x = 0 to x = 1 * f 1 dx(log x) m n having extended the. Leonhard Euler THEOREM 1 PROOF.

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1 Epasio of the itegral f dlog havig eteded the itegral fro the value = 0 to = * Leohard Euler THEOREM If deotes a positive iteger ad the itegral f d g is eteded fro the value = 0 to = the value of the itegral will be = g f f + g f + g f + g f + g. PROOF It is kow that the itegral f d g i geeral ca be reduced to this oe f d g sice it is possible to defie costat quatities A ad B i such a way that *Origial title: Evolutio forulae itegralis f dlog itegratioe a valore = 0 ad = etesa first published i Novi Coetarii acadeiae scietiaru Petropolitaae pp. 9-9 reprited i Opera Oia: Series Volue 7 pp. 6-7 Eeströ-Nuber E traslated by: Aleader Aycock for the project Euler-Kreis Maiz

2 f d g = A f d g + B f g ; for by differetiatio this equatio results f d g = A f d g + B f f d g Bg f +g d g which divided by f d g gives g = A + B f g Bg g or g = A Bg + B f + g g ; i order for this equatio to hold it is ecessary that whece we coclude = B f + g ad A = Bg B = f + g ad A = g f + g. Therefore we will have the followig geeral reductio f d g = g f + g f d g + f + g f g ; because it vaishes for = 0 if f > 0 of course the additio of a costat is ot ecessary. Hece havig eteded both itegrals to = the last absolute part vaishes by itself ad for the case = it will be Sice for = it is f d g = g f + g f d g 0 = f f = f f d g. havig put = we obtai the followig values for the sae case =

3 f d g = g f f d g = g f f d g = g f f + g f + g f + g f + g f + g f + g ad hece we coclude that for ay positive iteger it will be f d g = g f f + g if oly the ubers f ad g are positive. f + g f + g f + g COROLLARY Hece vice versa the value of a product of this kid fored fro a arbitrary aout of factors ca be epressed by a itegral so that f + g f + g f + g f + g = f g havig eteded this itegral fro the value = 0 to =. f d g COROLLARY Therefore if oe cosiders a progressio of this kid f + g f + g f + g f + g f + g f + g f + g f + g f + g f + g etc. its geeral ter correspodig to the idefiite ide is coveietly represeted by this itegral f g f d g ; ad usig this forula the progressio ad its ters correspodig to fractioal idices ca be ehibited. COROLLARY If we write istead of we will have

4 f + g f + g f + g f + g = f g f d g which equatio ultiplied by f +g yields f + g f + g f + g f + g = f g g f + g f d g. SCHOLIUM It would have bee possible to derive this last forula iediately fro the precedig oe sice we just proved that f d g = g f + g f d g if both itegrals are eteded fro the value = 0 to = ; this is to be kept i id for all the itegrals i everythig that follows. Furtherore it is to be oted that the quatities f ad g are positive which coditio was used i the proof of course. Cocerig the uber if it deotes the ide of a certai ter of the progressio that ide ca also be egative because all ters also those correspodig to egative idices of the progressio are cosidered to be ehibited by the give itegral forula. Nevertheless it is to be oted that this reductio f d g = g f + g f d g is oly true if > 0 because otherwise the algebraic part f +g f g would o vaish for =. SCHOLIUM 6 I already studied series of this kid which ca be called trascedetal because the ters correspodig to fractioal idices are trascedetal quatities i Coet. acad. sc. Petrop. book i ore detail ; therefore I will ot ivestigate those progressios here agai but focus o the rearkable Euler refers to his paper De progressioibus trascedetibus seu quaru terii geerales algebraice dari equeut. This is paper E9 i the Eeströ-Ide

5 coparisos of the itegral forulas that ca be derived fro it. After I had show that the value of the idefiite product is epressed by the itegral forula d log eteded fro = 0 to = what if is a positive iteger is aifest by direct itegratio I eaied the cases i which a fractioal uber is take for ; i these cases it is ideed ot obvious at all to which kid of trascedetal quatities these ters are to be referred. But by a sigular artifice I reduced the sae ters to better-kow quadratures; therefore this sees to be worth oe s while to cosider it with all eageress. 7 Sice it was deostrated that it is PROBLEM f + g f + g f + g f + g = f g f d g havig eteded the itegral fro = 0 to = to assig the value of the sae product i the case g = 0 by eas of a itegral. SOLUTION Havig put g = 0 i the itegral the ter g vaishes but at the sae tie also the deoiator g vaishes whece the questio reduces to the task to defie the value of the fractio g g i the case g = 0 i which so the uerator as the deoiator vaishes. Hece let us cosider g as a ifiitely sall quatity ad because it is g = e g log it will be g = + g log ad hece g = g log = g log ; hece our itegral becoes f f d log for this case so that oe ow has this epressio or f = f = f + f d log f d log.

6 COROLLARY 8 If is a positive iteger the itegratio of the itegral f d log succeeds ad havig eteded it fro = 0 to = ideed the product we foud to be equal to it results. But if fractioal ubers are take for the sae forula ca be applied to iterpolate this hypergeoetric progressio or etc etc. COROLLARY 9 If the epressio just foud is divided by the pricipal oe a product whose factors proceed i a arithetic progressio will eerge aely f + g f + g f + g f + g = f g f d log f d g whose values ca also be assiged if is a fractioal uber usig the itegral. 0 Because it is f d g = COROLLARY g f + g i like aer for the case g = 0 it will be f d log = f ad hece by those other itegrals ad = f f d g f d log f d log 6

7 f + g f + g f + g = f g f d log f + g f d g. SCHOLIUM Because we foud that = f + f d log it is plai that this itegral does ot deped o the value of the quatity f what is also easily see by puttig f = y whece we first fid f f d = dy ad log = log = f log y = f log y ad therefore f log = log y so that = dy log y which epressio results fro the first by puttig f =. Therefore for a iterpolatio of this kid the whole task is reduced to defiitio of the values of the itegral d log for the cases i which the epoet is a fractioal uber. For eaple if it is = oe has to assig the value of the forula d log which value I already oce showed to be = π while π deotes the circuferece of the circle whose diaeter is = ; but for other fractioal ubers I taught how to reduce its value to quadratures of algebraic curves of higher order. Because this reductio is by o eas obvious ad is oly valid if the itegratio of the forula d log is eteded fro the value = 0 to = it is sees to be worth oe s attetio. But eve though I already treated this subject oce I evertheless because I was led to the results i a rather o straight-forward way decided take o this subject here agai ad eplai everythig i ore detail. Euler cosidered this epressio also i E9 etioed already i the footote above. 7

8 THEOREM If the itegrals are eteded fro the value = 0 to = ad deotes a positive iteger it will be = g f +g d g f d g f d g whatever positive ubers are take for f ad g. Because above we showed that PROOF f + g f + g f + g = f g g f + g f d g if we write istead of we will have f + g f + g f + g = f g g f + g f d g. Now divide the first equatio by the secod ad this third oe will result f + + g f + + g f + g + + = g f + g f f + g d g f d g. But if oe writes f + g istead of f i the first equatio this fourth equatio will result f + + g f + + g f + g = f + gg g f + g f +g d g. Multiply this fourth equatio by the third ad oe will fid the equatio to be deostrated aely = g f +g d g f d g f d g. 8

9 COROLLARY If oe sets f = ad g = i the first equatio the sae product will result of course + + = d g ; havig copared this equatio to the oe etioed above we obtai d g f f +g d g = d g f d g. COROLLARY If we write g istead of i that equatio it will be + + = g g d g so that we fid this copariso of the followig itegral forulas g d g = f +g d g f d g f d g. COROLLARY If we put g = 0 i the equatio of the theore because of g = g log the powers of g will cacel each other ad this equatio will result + + = f d log f d log f d log whece we coclude f d log f d log = g g d g or because of 9

10 f d log = f f d log this oe f f d log f d log = g g d g. COROLLARY 6 Let us put f = g = ad = here so that is a positive iteger ad because of it will be ad hece d log = d log = d log = ad by takig = because of d = π d d oe will have d log = d = π. SCHOLIUM 7 So lo ad behold this succict proof of the theore I oce propouded ad statig that d log = π ad ote that I did ot use a arguet 0

11 ivolvig iterpolatios which I had used back the. Here it was of course deduced fro this theore I foud here ad which states that f d log f d log = g g d g. But the pricipal theore whece this oe is deduced reads as follows g f d g f +g d g = f d g d ; for each side if it is actually calculated by a itegratio eteded fro = 0 to = is equal to this product + +. But if we wat to give the oe side a ore geeral for ivolvig a furtheretedig class of itegrals we ca state the theore i such a way that g f d g f +g d g f d g = k k d k ; ad if here oe takes g = 0 it is f d log f d log = k k d k. Therefore it is especially to be oted that that equality holds whatever ubers are take for f ad g; i the case f = g this is ideed clear because it is for it will be ad because g g d g = g g+g d g = k g = g ; k d k

12 g+g d g = g d g the equality is obvious because k ca be take arbitrarily. But i the sae way we got to this theore it is possible to get to other siilar oes. THEOREM 8 If the followig itegrals are eteded fro the value = 0 to = ad deotes ay positive iteger it will be + + = g f +g d g f d g f d g whatever positive ubers are take for f ad g. PROOF I the precedig theore we already saw that f + g f + g f + g = f g g f + g f d g ; if i like aer we write istead of i the pricipal forula we will have f + g f + g f + g = f g g f + g f d g ; hece dividig this equatio by the first oe is led to f + + g f + + g f + g + + = g f + g f f + g d g f d g. But if we write f + g istead of f i the pricipal equatio we obtai this equatio

13 f + + g f + + g f + g = f + gg g f + g f +g d g. Now ultiply this equatio by the precedig ad the equatio to be proved will result + + = g f +g d g f d g f d g. COROLLARY 9 We obtai the sae value fro the pricipal equatio by puttig f = ad g = so that + + = d which itegral forula by writig k istead of is trasfored ito this oe so that k k d k g f +g d g f d log f d log = k k d k. COROLLARY 0 If we set g = 0 here because of g = g log equatio we will have this f d log f d log f d log = k k d k ; because we had foud before that

14 f d log f d log = k k d k by ultiplyig both epressios by each other we will have this equatio f d log f d log = k k d k k d k. COROLLARY Without ay restrictio oe ca put f = here; for the for = ad k = it will be ad because of d log d log 0 = 9 d d d log = d log ad d log 0 = d log = d d ; but the for = ad k = it will be or d log d log = 9 d d d log = d d.

15 GENERAL THEOREM If the followig itegrals are eteded fro the value = 0 to = ad deotes a positive iteger it will be λ + λ + λ + = λ λ + g whatever positive ubers are take for the letters f ad g. f +λg d g f d g λ f d g λ+ Because as we showed above it is PROOF f + g f + g f + g = f g g f + g f d g if we write λ istead of here at first but the λ + istead of we will obtai these two equatios λ f + g f + g f + λg = f λg g λ f + λg λ + f + g f + g f + λ + g = f λ + g g λ+ f + λ + g dividig the first equatio by this oe gives f + λg + g f + λg + g f + λg + g λ + λ + λ + f d g λ f d g λ+ ; λ f + λg + g f = g λ + f + λg d g λ f d g λ+. But if we write f + λg istead of f i the first equatio we will obtai f + λg + g f + λg + g f + λg + g = f + λgg g f + λg + g f +λg d g which two equatios ultiplied by each other produce the equatio to be deostrated λ + λ + λ + = λg f +λg d g f d g λ λ + f d g λ+.

16 COROLLARY If we put f = λ ad g = i the pricipal equatio we will also fid λ + λ + λ + = λ λ + which for writig k istead of chages ito this oe λk λ + λk d k so that we have this very far-etedig theore λ d g f +λg d g f d g λ f d g λ+ = k λk d k. COROLLARY This theore ow holds eve if is ot a iteger; let us because the uber λ ca be take arbitrarily eve write istead of λ ad we will fid this theore f d g f d g + = k k d k g f +g d g. COROLLARY If we put g = 0 because of g = g log for that theore will take this f d log f d log + = k k d k f d log which is ore coveietly represeted as follows f d log f d log f d log + = k k d k ; here it is evidet that the ubers ad ca be peruted. 6

17 SCHOLIUM 6 So we foud two ways how ay coparisos ad relatios of itegrals forulas ca be derived; the oe way foud i cotais itegrals of this kid p d g q which I already treated soe tie ago i y observatios o the itegrals p d q eteded fro the value = 0 to = ; there I showed at first that the letters p ad q ca be iterchaged that it is p d q = q d p but the that p d p = π si pπ ; But especially I deostrated so that p d q p+q d = r p d r p+r d ; q the copariso foud i is already cotaied i this equatio so that othig ew I have ot already eplaied ca be deduced fro this. Therefore here I aily attept to follow the other way eplaied i ; sice without ay restrictio oe ca take f = our priary equatio will be d log d log d log + = k k d k by eas of which the values of the itegral forula d log λ if λ is ot a iteger ca be reduced to quadratures of algebraic curves; sice if λ is a iteger the itegrals ca be solved eplicitly because it is Euler agai refers to his paper Observatioes circa itegralia forularu p d q posito post itegratioe =. This is paper E i the Eeströ-Ide. 7

18 d log λ = λ. But the questio of greatest iportace cocers the cases i which λ is a ratioal uber. Therefore I will defie these here successively for soe sall deoiators. PROBLEM 7 While i deotes a positive iteger to defie the value of the itegral d log i havig eteded the itegratio fro = 0 to =. SOLUTION Let us put = i our geeral equatio ad it will be d log d log = k k d k. Now let it be = i ad because of = i + it will be d log = i + ; ow further take k = so that k = i + ad it will be ad hece log d i i + = log d i = i + i+ d i i+ d i where it is evidet to be sufficiet to take oly odd ubers for i because for the eve oes the epasio is obvious iediately. 8

19 COROLLARY 8 But all cases are easily reduced to i = or eve to i = ; for if i + is ot a egative uber the reductio we foud holds. For this case it will therefore be because of d = π. d log = d = π COROLLARY 9 But havig covered these pricipal cases because of we will have ad i geeral d log = d log = π d log + d log d log = π = 7 + π. PROBLEM 0 While i deotes a positive iteger to defie the value of the itegral d log i havig eteded the itegratio fro = 0 to =. SOLUTION Let us start fro the equatio of the precedig proble d log d log = k k d k 9

20 ad let us put = i the geeral forula so that oe has d log d log d log = k k d k ad by ultiplyig these two equatios we obtai d log d log = kk k d k k d k. Now just put = i here that d log i = i ad take k = ad this equatio will result log d i i = 9 i d i i d i whece we coclude log d i = 9 i i d i i d. i COROLLARY Here two pricipal cases occur o which all reaiig oes deped aely the cases i = ad i = ; for these cases it is I. II. d log = 9 d log = 9 d d d d ; 0

21 the last forula because of d = d ca be trasfored ito this oe d log d = d COROLLARY If for the sake of brevity as i y observatios etioed before we put p d p = q q ad as we did it there for this class also set = π si π = α but the put it will be = d = A I. II. d log = 9 d log = = 9αA αα = A. Euler refers to his paper E agai.

22 COROLLARY Therefore for the first case we will have d log = 9αA d log = 9αA ad d log + = 7 + 9αA but for the other case o the other had we will fid d log = αα A d log = αα A ad d log = 8 αα A. PROBLEM While i deotes a positive iteger to defie the value of the itegral d log i havig eteded the itegratio fro = 0 to =. SOLUTION I the solutio of the precedig proble we were led to this equatio d log d log = kk k d k but the geeral forula by settig = i it yields k d k ; d log d log d log = k by cobiig these forulas we obtai k d k ;

23 d log d log = k k d k Let = i ad take k = ad it will be k d k k d k. i d log = i i d i i d i i d. i COROLLARY So if i = we will have this equatio d log = d d d ; if this epressio is deoted by the letter P it will be i geeral d log = 9 P. COROLLARY 6 For the other pricipal case let us take i = ad it will be d log = d d 8 d or after soe siplificatio d log = 8 d d d ; if this epressio is deoted by the letter Q it will be i geeral d log = 7 Q.

24 SCHOLIUM 7 If we idicate the itegral forula p d by the sig q solutio i geeral will be as follows d log i i = i i ad for the two cases epaded before P = ad Q = 8 Now for the forulas depedig o the circle let us put = π si π = α ad = but for trascedetal oes of higher order let = d = π si π i i d = A o which all reaiig oes deped; hece we will fid whece it is clear that But because α = P = αα AA ad Q = ααβ β AA PQ = α = π si π. π ad β = π it will be i i. = β P = π πaa ad Q = 8AA ad P Q = A. π PROBLEM p q the 8 While i deotes a positive iteger to defie the value of the itegral d log i havig eteded the itegratio fro = 0 to =.

25 SOLUTION Fro the precedig solutios it is already perspicuous that for this case oe will obtai this forula at the ed i d log = i i d i i d i i d i i d i which itegral forulas belog to the fifth class itroduced i y dissertatio etioed above. Hece if i the sae way as it was doe there the sig p q deotes this forula p d the value i questio ca be ore q coveietly epressed i such a way that d log i = i i i i i i i i ; i here it ideed suffices to have assiged values saller tha five to i; for if the uerators eceed five it is just ote that but the further i + i i = + i = + i = + i i + + i i + i i + 0 i Furtherore for this class two forulas ideed ivolve the quadrature of the circle; these forulas are = π si π = α ad = π si π. = β but the two cotai higher quadratures which we wat to put Euler agai refers to his paper Observatioes circa itegralia forularu p d q posito post itegratioe =. This is paper E i the Eeströ-Ide.

26 = d = d = A ad = d = B ad usig these I assiged the values of all reaiig forulas of this class 6 aely = = = = α = β A = β B = A = β = ββ αb ; = αb β = B; = αa β. = = ; = α A ; COROLLARY Havig take the epoet i = it will be d log = = α β A B whece we coclude i geeral while deotes a positive iteger that d log = 6 α β A B. COROLLARY 0 Now let i = ad sice the this equatio results d log 6 8 = 6 Euler took the followig list out of E. 6

27 because of 6 = = ad 8 = the left-had side will be ad i geeral = αβ BB A d log = 7 αβ BB A. Let i = ad the for foud COROLLARY d log = 6 9 because of 6 = 9 = 7 = 7 0 chages to = β α A BB whece it is cocluded that i geeral d log = 8 β α A BB. 7

28 Fially for i = our equatio because of d log COROLLARY = = 7 = = 6 0 will be trasfored ito this for so that i geeral 6 d log = ααββ AAB = 9 ααββ AAB. SCHOLIUM If we represet the value of the itegral forula d log λ by the sig λ] the cases epaded up to ow yield ] = α β A B ] = αβ BB A ] = β β ] = α β A BB AAB + ] = α β A B + ] = αβ BB A + ] = β α A BB + ] = α β AAB 8

29 whece by cobiig two whose idices add up to 0 we coclude + ] ] = α = π si π + ] ] = β = π si π + ] ] = β = π si π + ] ] π = α = si π. But fro the precedig proble i like aer we deduce: ] = P = αα ] = Q = ααβ β AA AA + ] = αα β AA + ] = ααβ AA ad hece + ] ] = α = π si π + ] ] = α = π si π whece i geeral we obtai this theore that λ] λ] = λπ si λπ ; the reaso for this ca be give fro the iterpolatio ethod eplaied soe tie ago 7 as follows. Because it is λ] = λ λ + λ λ λ + λ λ λ + λ etc. 7 Euler eplais the iterpolatio ethod he talks about here also i E9. 9

30 it will be ad hece λ] = +λ λ λ +λ λ λ +λ λ λ etc. λ] λ] = λλ as I deostrated elsewhere 8. λλ λπ etc. = 9 λλ si λπ PROBLEM 6 - GENERAL PROBLEM If the letters i ad deote positive itegers to defie the value of the itegral d log i or d log i havig eteded the itegratio fro = 0 to =. SOLUTION The ethod up to ow will ehibit the value i questio epressed by quadratures of algebraic curves i the followig way log d i = i i d i i d i i d i Hece if for the sake of brevity we deote the itegral forula p d q p by this character q but o the other had the forula log d by this character ] so that ] deotes the value of this idefiite product z while z = the i value questio will epressed ore succictly as follows 8 Euler proved this relatio i his paper Methodus facilis coputadi aguloru sius ac tagetes ta aturales qua artificiales. This is paper E8 i the Eeströ-Ide. 0

31 ] i i = i i whece it is also cocluded ] i = i i i i i i i i i i i i i i i i i Here it will always suffice to have take the uber i saller tha because it is kow for larger ubers that i. ] i + = i + ] i i the sae way ] i + = i + i + ] i etc. ad so the whole ivestigatio is hece reduced to those cases i which the uerator i of the fractio i is saller tha the deoiator. I additio it will be helpful to have oted the followig properties of the itegral forulas p d p = : q q I. The letters p ad q are iterchageable so that p = q q. p II. If oe of the two ubers p or q is equal to the epoet the value of the itegral forula will be algebraic aely p = = p p or q = = q q. III. If the su of the ubers p + q is equal to the epoet the value of p the itegral forula q ca be ehibited by eas of the quadrature of the circle because p = p p = p π si pπ ad q = q q = q π si qπ.

32 IV. If oe of the ubers p or q is greater tha the epoet the itegral p forula q ca be reduced to aother oe whose ters are saller tha ; this is achieved usig this reductio p + q = p p + q p. q V. There is a relatio aog ay of these itegral forulas of such a kid that p p + q p = q r r p + r q = q r q + r ; p by eas of this relatio all reductios I gave i y observatios o these forulas 9 are foud. COROLLARY If we accoodate the foud forula to the sigle cases this way by eas of reductio IV we will be able to ehibit the i the ost siple way as follows. Ad for the case = i which o further reductio is ecessary we will have ] = = π si π = π. COROLLARY 6 For the cases = we will have these reductios ] = ] =. 9 Euler is agai referrig to E.

33 COROLLARY 7 For the case = oe obtais these three reductios because of = ] = ] = ] = = because i the secod equatio it is before be ] = ; = ] = π. COROLLARY 8 Now let = ad these four reductios result ] = ] = ] = ] = = π it will of course as.

34 COROLLARY 9 Let = 6 ad we will have these reductios ] = ] = 6 6 ] = ] = ] = = 6 = = 6 6. COROLLARY 6 0 For = 7 the followig si equatios result ] = ] = ] = ] = ] = ] 6 =

35 COROLLARY 7 Now let = 8 ad oe will get to these seve reductios ] = ] = = 6 8 ] = ] = = 8 ] = ] 6 = = ] 7 = SCHOLIUM It would be superfluous to epad these cases ay further because the structure of these forulas is already see very clearly fro the oes give. If the ubers ad are coprie i the propouded forula ] the rule is obvious because ] = but if these ubers ad have a coo divisor it will ideed be useful to reduce this fractio to the sallest for ad etract the value i questio fro the precedig cases; evertheless the operatio ca also be doe as follows. Because the epressio i questio certaily has this for ;

36 ] = PQ where Q is the product of the itegral forulas P o the other had the product of soe absolute ubers i order to fid that product Q just cotiue this series of forulas etc. util the uerator eceeds the epoet ad istead of this uerator write its ecess over ; if this ecess is put = α so that our forula is α this uerator α will give a factor of a product P; the cotiue this series of forulas α α+ α+ etc. util oe agai gets to a uerator greater tha the epoet ad the β forula eerges; istead of this forula oe the has to write +β ad hece the factor β is itroduced ito the product ad oe has to cotiue like this util forulas for Q will have eerged. To uderstad these operatios ore easily let us epad the case of the forula ] 9 = 9 PQ this way; the letters P ad Q are foud as follows: for Q for P ad so oe fids 9 6 Q = ad P = Because it is 9 = 9 it is PQ = ad hece ] 9 = THEOREM Whatever positive ubers are idicated by the letters ad i the otatio itroduced ad eplaied before it will always be 6

37 ] =. PROOF For the cases i which ad are coprie ubers the validity of this theore was show i the precedig theores; but that it also holds if those ubers ad have a coo divisor is ot evidet fro that theore; but sice the forula was already proved to be true i the cases i which ad are utually prie it is atural to coclude that this theore is true i geeral. I a copletely aware that this kid to deduce soethig is copletely uusual ad has to see suspect to ost people. I order to clear those doubts because for the cases i which the ubers ad are coposite we obtaied two epressios it will be useful to have show the agreeet for the cases eplaied before. Ad the case = is already a huge cofiratio i which case our forula obviously becoes =. COROLLARY The first case requirig a deostratio of the agreeet is that oe i which it is = ad = for which we foud above 7 but ow via the theore it is where by copariso it is ] = ; ] = = whose validity was cofired i y observatios etioed 0 above. 0 Euler agai refers to his paper Observatioes circa itegralia forularu p d q posito post itegratioe =. This is paper E i the Eeströ-Ide. 7

38 COROLLARY If it is = ad = 6 usig the results derived above 9 we have ] = ; ow o the other had by eas of the theore ] = ad therefore it has to be = whose validity is clear for the sae reasos. COROLLARY 6 If it is = ad = 6 oe gets to this equatio = but if = ad = 6 i like aer or = = which is also detected to be true. ; COROLLARY 7 The case = ad = 8 yields this equality 6 = 7 8

39 but the case = ad = 8 this oe = 6 ad fially the case = 6 ad = 8 gives this equatio 6 6 which is also true. 6 = SCHOLIUM 8 But if i geeral the ubers ad have the coo factor ad the propouded forula is ] = ] because ] = after havig reduced the sae to the epoet it will be 6 6. By the theore the sae epressio o the other had becoes whece for the epoet it will be 6 6 = If i the sae way the coo divisor is for the epoet oe will fid. 9

40 = which equatio ca be ore coveietly ehibited as follows = 6 7 But if i geeral the coo divisor is d ad the epoet d oe will have d d d d d d d d d d d d = d d d d d d d d which equatio ca easily be accoodated to ay cases whece the followig theore deserves it to be oted. d. THEOREM p 9 If α was a coo divisor of the ubers ad ad the forula q deotes the value of the itegral p d eteded fro = 0 to = it will q be α α α α α α α = α α. PROOF The validity of this theore is already see fro the precedig Scholiu; while the coo divisor was = d ad the two propouded ubers were d ad d there here I just wrote ad istead of the but istead of 0

41 their divisor d I wrote the letter α which kid of divisor the stated equality cotais i such a way that oe assues the ubers ad ad hece also α ad α to occur i the cotiued arithetic progressio α α α etc. I additio I a forced to cofess that this deostratio is of course aily based o iductio ad caot be cosidered to be rigorous by ay eas; but because we are evertheless coviced of its truth this theore sees to be worth oe s greater attetio; there is evertheless o doubt that a further epasio of itegral forulas of this kid will fially lead to a coplete proof; but it is a etraordiary specie of aalytical ivestigatio that it was possible for us to see its truth before the coplete proof. COROLLARY 60 So if we substitute the itegrals for the sigs we itroduced our theore will be as follows α α α α α d α d α d = α d d d. COROLLARY 6 Or if for the sake of brevity we set = X it will be α α α α = α α d X d X α d X d X α d X d X d X.

42 GENERAL THEOREM 6 If the divisors of the two ubers ad are α β γ etc. ad the forula p q deotes the value of the itegral p d eteded fro = 0 to = q the followig epressios cosistig of itegral forulas of this kid will be equal to each other α α α α α α α α α β β β β β = β β β β γ γ γ γ γ = γ γ γ γ etc. PROOF The validity of this theore obviously follows fro the precedig theore because every sigle oe of these epressios is equal to this oe which correspods to the uity as sallest coo divisor of the ubers ad. Therefore so ay epressios of this kid all equal to each other ca be ehibited as there were coo divisors of the two ubers ad. COROLLARY 6 Because this forula is = ad hece = our equal epressios ca represeted be ore succictly as follows

43 α α α α α α β β β = β β β γ γ γ = γ γ γ α β γ. For eve if the uber of factors was icreased here the structure of these forulas is evertheless easily see. COROLLARY 6 So if = 6 ad = because of the coo divisors of these ubers 6 oe will have the followig four fors that are all equal to each other = = = = COROLLARY 6 If the last forula is cobied with the peultiate this equatio will result = but the last copared to the secod yields =

44 SCHOLIUM 66 Hece ifiitely ay relatios aog the itegral forulas of the for p d p = q q follow which are eve ore rearkable because we were led to the by a copletely sigular ethod. Ad if ayoe does ot believe the to be true he or she should cosult y observatios o these itegral forulas ad will the hece easily be coviced of their truth for ay case. But eve if this cosideratio provides soe cofiratio the relatios foud here are evertheless of eve greater iportace because a certai structure is oticed i the ad they are easily geeralized to all classes whatever uber was assued for the epoet whereas i the first treatet the calculatio for the higher classes becoes cotiuously ore cubersoe ad itricate. Euler agai refers to his paper Öbservatioes circa itegralia forularu p d q posito post itegratioe = ". This is paper E i the Eeströ-Ide.

45 SUPPLEMENT CONTAINING THE PROOF OF THE THEOREM PROPOUNDED IN It is coveiet to derive this proof fro the results etioed above; just take the equatio give i which for f = havig chaged the letters reads as follows d log ν d log µ d log ν+µ = κ κµ d κ ν ad usig kow reductios represet it i this for d log ν d log µ d log ν+µ = κµν µ + ν κµ d κ ν. Now set ν = ad µ = λ but the κ = so that we have d log d log λ = λ λ d λ+ d log λ + κ which for the sake of brevity havig used otatio itroduced above is ore coveietly epressed as follows ] λ ] λ+ ] = λ λ. λ + Now successively write the ubers... istead of λ ad ultiply all these equatios whose uber is = ad the resultig equatio will be = + ] ] ] ] ] + ] + ] + ] + ] = But i like aer just trasfor the left-had side so that ] + ] ] ] ] ] + ] + ] + ].

46 whose agreeet with the precedig is revealed by cross ultiplicatio. But because fro the ature of these forulas ] + = + ] ] + = + ] ] + = + ] etc. ad sice we have of these forulas here this left-had side will becoe ] ; because this oe is equal to the other part ehibited before aely we obtai this equatio so that ] = ] = ; because this equatio = agrees with the oe propouded i its truth is ow ideed proved fro ost certai priciples. PROOF OF THE THEOREM PROPOUNDED IN 9 Also this theore eeds a ore rigorous proof I will give usig the equatio established before aely ] λ ] λ+ ] = λ λ λ + as follows. While α is a coo divisor of the ubers ad successively write the ubers α α α etc. up to istead of λ whose total aout is = α ad ow ultiply all equatios resultig this way so that this equatio eerges 6

47 = α α + α ] α ] α ] α ] α ] +α ] +α ] +α ] + ] α + α α + α + α α Now trasfor the left-had side ito this oe equal to it. ] α ] α ] α ] α ] +α ] +α ] +α ] + ] which because of ] +α = +α α ] ad so for the reaiig oes is reduced to this oe ] α + α + α + α + The right-had side of the equatio i like aer is trasfored ito this oe α α + α α + α whece this equatio results ad hece α + α α + ] α α = α α α α α ] = α α α α α α α α α α α which equatio copared to the precedig yields this equatio α α α α α α α = what is to be uderstood for all coo divisors of the two ubers ad. 7