```Date: Feb 24, 2013 10:11 AM
Author: clicliclic@freenet.de
Subject: An independent integration test suite

The Russian book "Integration of Functions"(?????????????? ???????) published by A.F. Timofeev(?.?. ????????) in 1948 provides many integration Examples(???????) of the following general kind:INT(LN(x)^2*x, x) = x^2/2*(LN(x)^2 - LN(x) + 1/2)INT(LN(x)/x^5, x) = - (4*LN(x) + 1)/(16*x^4)INT(x^2*LN((x - 1)/x), x) = x^3/3*LN((x - 1)/x) - LN(x - 1)/3 - x*(x +2)/6INT(COS(x)^5, x) = SIN(x)/15*(3*COS(x)^4 + 4*COS(x)^2 + 8)INT(SIN(x)^2*COS(x)^4, x) = 1/6*SIN(x)^3*COS(x)^3 + 1/8*SIN(x)^3*COS(x)- 1/16*SIN(x)*COS(x) + x/16INT(1/SIN(x)^5, x) = 3/8*LN(TAN(x/2)) - 3*COS(x)/(8*SIN(x)^2) -COS(x)/(4*SIN(x)^4)INT(SIN(x)/#e^x, x) = - (COS(x) + SIN(x))/(2*#e^x)INT(#e^(2*x)*SIN(3*x), x) = 1/13*#e^(2*x)*(2*SIN(3*x) - 3*COS(3*x))INT(a^x*COS(x), x) = a^x/(LN(a)^2 + 1)*(LN(a)*COS(x) + SIN(x))INT(COS(LN(x)), x) = x/2*(COS(LN(x)) + SIN(LN(x)))INT(SEC(x)^2*LN(COS(x)), x) = TAN(x)*LN(COS(x)) + TAN(x) - xINT(x*TAN(x)^2, x) = LN(COS(x)) + x*TAN(x) - 1/2*x^2INT(ASIN(x)/x^2, x) = - ASIN(x)/x + LN((1 - SQRT(1 - x^2))/x)INT(ASIN(x)^2, x) = x*ASIN(x)^2 + 2*SQRT(1 - x^2)*ASIN(x) - 2*xINT(x^2*ATAN(x)/(1 + x^2), x) = x*ATAN(x) - 1/2*ATAN(x)^2 - 1/2*LN(x^2 +1)INT(ACOS(SQRT(x/(x + 1))), x) = (x + 1)*(ACOS(SQRT(x/(x + 1))) +SQRT(1/(x + 1))*SQRT(x/(x + 1)))There are 8 chapters in the book containing 81+90+14+132+120+26+11+59 =533 integration examples; the above are numbers 76 to 81 from chapter 1.I believe the whole would make a good independent integration test suitebecause the book antedates all automated integrators and because theauthor claims to cover the field thoroughly (with respect to integralsexpressible by elementary functions and by elliptic integrals). In theForeword (???????????), he writes (as translated by Google):"In most of the guides on higher mathematics the question of integrationof functions of one independent variable has a fairly complete coverage,so that very often students do not get a clear idea of ??what functionsare integrated in closed form, for which this integration is impossible,and what techniques are useful in a particular case for various kinds offunctions. With this in mind, the author of this book sought to explainthe issue to a possible full, paying particular attention to thepractice of integration, thus introducing a large number of examples.Thus, this book can serve as a first, a reference book for those wantingto get a quick response with respect to a quadrature, and secondly, aidfor students who wish to supplement and deepen their knowledge in thismatter."Some may prefer the original:"? ??????? ????? ?????????? ?? ?????? ?????????? ?????? ?? ????????????????????? ?????? ???????????? ??????????? ?? ????? ?????????? ????????????????, ?????????? ???? ????? ????????????? ?? ???????? ??????????????????? ? ???, ????? ??????? ????????????? ? ???????? ????, ???????? ??? ?????????????? ?????????? ?????? ?????? ?????????????????????? ? ??? ??? ???? ?????? ??? ????????? ????? ???????. ???? ??? ?????, ????? ? ????????? ????? ????????????????? ?????? ? ?????????????????, ??????? ?????? ???????? ?? ???????? ??????????????, ????? ??????? ??????? ?????????? ????????. ????? ???????, ????? ??? ????????????, ??-??????, ???????????? ??? ???, ???????? ???????? ?????? ????????????????? ??? ??? ???? ??????????, ???-??????, ???????? ???????????, ???????? ????????? ? ???????? ???? ?????? ? ???? ???????."A djvu file of the book (6 Megabyte) can be downloaded freely fromvarious websites - just google the Russian title. The full name of theauthor apparently was Aleksandr Fedotovich Timofeev (?????????????????? ????????), but I couldn't locate anyinformation on him beyond this. A Russian review of the book isavailable at:<http://www.mathnet.ru/links/5566c95518efdede979ffdc8a2278b01/rm8600.pdf>There are the usual misprints in the book, but with both the integrandand antiderivative available, the original meaning can be reconstructedfor all (or almost all) examples I think. Timofeev's antiderivatives areusually close to the most compact form possible (an exception is hisconsistent use of logarithms for inverse hyperbolic functions), but theyhave to be checked for validity over the complex plane, and be correctedif necessary (this seems rare, but was needed for the last exampleabove). Apart from compactness, continuity (and reality) on the realaxis might be worth checking and repairing too.So, if 5 to 10 people were found willing to type in and check (andperhaps correct or improve) 50 to 100 integrals and evaluations each, adigitization of this corpus could be an almost pleasant task, and surelyquite useful. What do you think?Martin.
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