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)/6

INT(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/16

INT(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) - x

INT(x*TAN(x)^2, x) = LN(COS(x)) + x*TAN(x) - 1/2*x^2

INT(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*x

INT(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 suite
because the book antedates all automated integrators and because the
author claims to cover the field thoroughly (with respect to integrals
expressible by elementary functions and by elliptic integrals). In the
Foreword (???????????), he writes (as translated by Google):

"In most of the guides on higher mathematics the question of integration
of functions of one independent variable has a fairly complete coverage,
so that very often students do not get a clear idea of ??what functions
are integrated in closed form, for which this integration is impossible,
and what techniques are useful in a particular case for various kinds of
functions. With this in mind, the author of this book sought to explain
the issue to a possible full, paying particular attention to the
practice of integration, thus introducing a large number of examples.
Thus, this book can serve as a first, a reference book for those wanting
to get a quick response with respect to a quadrature, and secondly, aid
for students who wish to supplement and deepen their knowledge in this
matter."

Some may prefer the original:

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????????? ?????? ?? ??????????????
??????? ?????? ???????????? ?????????
?? ?? ????? ?????????? ???????
?????????, ?????????? ???? ????? ?????
???????? ?? ???????? ??????
????????????? ? ???, ????? ??????? ????
????????? ? ???????? ????, ???
????? ??? ?????????????? ?????????? ?
????? ?????? ?????????????
????????? ? ??? ??? ???? ?????? ??? ???
?????? ????? ???????. ???? ??? ?
????, ????? ? ????????? ????? ?????????
???????? ?????? ? ?????????
????????, ??????? ?????? ???????? ?? ??
?????? ??????????????, ????? ???
???? ??????? ?????????? ????????. ????
? ???????, ????? ??? ?????
???????, ??-??????, ???????????? ??? ??
?, ???????? ???????? ?????? ?????
???????????? ??? ??? ???? ??????????, ?
??-??????, ???????? ???
????????, ???????? ????????? ? ???????
? ???? ?????? ? ???? ???????."

A djvu file of the book (6 Megabyte) can be downloaded freely from
various websites - just google the Russian title. The full name of the
author apparently was Aleksandr Fedotovich Timofeev (?????????
????????? ????????), but I couldn't locate any
information on him beyond this. A Russian review of the book is
available at:

<http://www.mathnet.ru/links/5566c95518efdede979ffdc8a2278b01/rm8600.pdf>

There are the usual misprints in the book, but with both the integrand
and antiderivative available, the original meaning can be reconstructed
for all (or almost all) examples I think. Timofeev's antiderivatives are
usually close to the most compact form possible (an exception is his
consistent use of logarithms for inverse hyperbolic functions), but they
have to be checked for validity over the complex plane, and be corrected
if necessary (this seems rare, but was needed for the last example
above). Apart from compactness, continuity (and reality) on the real
axis might be worth checking and repairing too.

So, if 5 to 10 people were found willing to type in and check (and
perhaps correct or improve) 50 to 100 integrals and evaluations each, a
digitization of this corpus could be an almost pleasant task, and surely
quite useful. What do you think?

Martin.