Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.



Re: integration test suite / Chap 3
Posted:
Apr 27, 2013 7:18 AM


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 nonexamples are represented by [].
Martin.
" Timofeev (1948) Ch. 3, examples 1  3 (p. 101) ... "
INT(1/((x2)^3*(x+1)^2),x)=(2*x^25*x1)/(18*(x+1)*(x2)^2)+1/27~ *LN((x2)/(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^42*x^3+27/2*x^2108*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))=(1/4SQRT(3)/4)*LN(2*xSQRT(3)+3)+(SQRT(3)/4+1/4)~ *LN(2*x+SQRT(3)+3)
INT((2*x3)/(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((x1)/(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*(76*x+2*x^2)^2),x)=1/(98*x^2)12/(343*x)+2*(419*x)~ /(1715*(76*x+2*x^2))+80/2401*LN(x)40/2401*LN(76*x+2*x^2)+234*~ SQRT(5)/60025*ATAN((2*x3)/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)*(110*(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((ab*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^1273*a^4*x^855*a^8*x^415*a^~ 12)/(768*a^4*(x^4+a^4)^4)+5/(256*a^6)*ATAN(x^2/a^2)
" ... end of Timofeev Ch. 3 "



