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Topic: integration test suite / Chap 3
Replies: 1   Last Post: Apr 27, 2013 7:18 AM

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clicliclic@freenet.de

Posts: 982
Registered: 4/26/08
Re: integration test suite / Chap 3
Posted: Apr 27, 2013 7:18 AM
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Axel Vogt schrieb:
>
> These are the excercises for Chap 3 in Timofeev's book,
> p.101 #1 - #3, p.105 #4 - #9, p.109 #10 - #12, p.113 #13, #14
>
> Two of them are 'reductions formula' for a linear term, #4 and #5.
> For those I have no idea how to test with Maple - they should be
> done by partial integration and thus I ignore them.
>
> The others seem to be for applying decomposition using partial
> fractions. For which I suppose, that all CAS do it. There seem
> to be no sophisticated cases there, so I just state _some_.
>


I have converted these to Derive and filled in the gaps. At least one
integrand is misprinted, as can be concluded from its antiderivative. My
evaluations stay close to Timofeev's ones, which are meant to reveal how
they have been arrived at, unless his expressions can be shortened
significantly. As predicted, Derive 6.10 has no problems here at all. My
file is appended; the two non-examples are represented by [].

Martin.


" Timofeev (1948) Ch. 3, examples 1 - 3 (p. 101) ... "

INT(1/((x-2)^3*(x+1)^2),x)=(2*x^2-5*x-1)/(18*(x+1)*(x-2)^2)+1/27~
*LN((x-2)/(x+1))

INT(1/((x+2)^3*(x+3)^4),x)=(60*x^4+630*x^3+2450*x^2+4175*x+2627)~
/(6*(x+2)^2*(x+3)^3)+10*LN((x+2)/(x+3))

INT(x^5/(3+x)^2,x)=1/4*x^4-2*x^3+27/2*x^2-108*x+243/(x+3)+405*LN~
(x+3)

" Timofeev (1948) Ch. 3, examples 4 - 9 (p. 105) ... "

[]

[]

INT(x/(3+6*x+2*x^2),x)=1/4*LN(-(3+6*x+2*x^2))+SQRT(3)/2*ATANH((3~
+2*x)/SQRT(3))=(1/4-SQRT(3)/4)*LN(2*x-SQRT(3)+3)+(SQRT(3)/4+1/4)~
*LN(2*x+SQRT(3)+3)

INT((2*x-3)/(3+6*x+2*x^2)^3,x)=-(8*x^3+36*x^2+44*x+13)/(4*(2*x^2~
+6*x+3)^2)+1/SQRT(3)*ATANH((3+2*x)/SQRT(3))

INT((x-1)/(x^2+5*x+4)^2,x)=(7*x+13)/(9*(x^2+5*x+4))+7/27*LN((x+1~
)/(x+4))

INT(1/(x^2+3*x+2)^5,x)=(2*x+3)/(4*(x^2+3*x+2)^4)*(-1+14/3*(x^2+3~
*x+2)-70/3*(x^2+3*x+2)^2+140*(x^2+3*x+2)^3)+70*LN((x+1)/(x+2))

" Timofeev (1948) Ch. 3, examples 10 - 12 (p. 109) ... "

INT(1/(x^3*(7-6*x+2*x^2)^2),x)=-1/(98*x^2)-12/(343*x)+2*(41-9*x)~
/(1715*(7-6*x+2*x^2))+80/2401*LN(x)-40/2401*LN(7-6*x+2*x^2)+234*~
SQRT(5)/60025*ATAN((2*x-3)/SQRT(5))

INT(x^9/(x^2+3*x+2)^5,x)=-(25*x^8+35292*x^7+369950*x^6+1632276*x~
^5+3919731*x^4+5527800*x^3+4578216*x^2+2063520*x+390960)/(24*(x^~
2+3*x+2)^4)+1472*LN(x+2)-1471*LN(x+1)

INT((1+2*x)^2/(3+5*x+2*x^2)^5,x)=-(11+10*x)/(4*(2*x^2+5*x+3)^4)+~
31*(5+4*x)/(6*(2*x^2+5*x+3)^3)*(1-10*(2*x^2+5*x+3)+120*(2*x^2+5*~
x+3)^2)+2480*LN((x+1)/(2*x+3))

" Timofeev (1948) Ch. 3, examples 13 - 14 (p. 113) ... "

INT((a-b*x^2)^3/x^7,x)=-a^3/(6*x^6)+3*a^2*b/(4*x^4)-3*a*b^2/(2*x~
^2)-b^3*LN(x)

INT(x^13/(a^4+x^4)^5,x)=x^2*(15*x^12-73*a^4*x^8-55*a^8*x^4-15*a^~
12)/(768*a^4*(x^4+a^4)^4)+5/(256*a^6)*ATAN(x^2/a^2)

" ... end of Timofeev Ch. 3 "



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