From: didier.oslo@hotmail.com   
      
   On Thursday, 7 November 2013 03:33:19 UTC+1, Nasser M. Abbasi  wrote:   
   > On 11/6/2013 6:16 PM, Albert Rich wrote:   
   >   
   > > I am currently transcribing the 109 integration examples in Chapter 8 of   
   Timofeev's book.   
   >   
   > >However I am unable to figure out what Timofeev intended for examples #66   
   and #69 on   
   >   
   > >page 366.  The integrands for #66 and #69 appear to be   
   >   
   > >   
   >   
   > > (cosh(x)^2 - sinh(x)^3) / (cosh[x)^3 + sinh(x)^3)   
   >   
   > >   
   >   
   > > and   
   >   
   > >   
   >   
   > > (tanh(x)-cosh(2x))*cosh(x)/((sinh(2x)+sinh(x)^2)*sqrt(sinh(2*x)))   
   >   
   > >   
   >   
   > > respectively, but the antiderivatives shown for them are not valid.   
   >   
   > >Can anybody out there help me out?   
   >   
   > >   
   >   
   > > Albert   
   >   
   > >   
   >   
   >   
   >   
   > Yes, there is an error/typo somewhere. difference between   
   >   
   > the derivative of the anti derivative and the integrand is  not   
   >   
   > zero and not even linear constant difference. This is from   
   >   
   > the 1948 edition. May be there is a newer edition than this   
   >   
   > to check?   
   >   
   >   
   >   
   > #66   
   >   
   > restart;   
   >   
   > integrand:=(cosh(x)^2 - sinh(x)^3) / (cosh(x)^3 + sinh(x)^3):   
   >   
   > anti:= 1/(3*(1+tanh(x)))+ 4/(3*sqrt(3))*arctan( (2*tanh(x)-1)/sqrt(3)):   
   >   
   > check:=simplify(integrand-diff(anti,x));   
   >   
   >   
   >   
   >   (1/3)*(3*cosh(x)^3-6*cosh(x)^2*sinh(x)+3*cosh(x)^2-4*cosh(x)   
   >   
   >   +2*sinh(x)) /(cosh(x)^3+cosh(x)^2*sinh(x)-sinh(x))   
   >   
   >   
   >   
   > plot(%,x=-Pi..Pi);   
   >   
   >   
   >   
   > #69   
   >   
   > restart;   
   >   
   > integrand:=(tanh(x)-cosh(2*x))*cosh(x)/((sinh(2*x)+sinh(x)^2)*   
   qrt(sinh(2*x))):   
   >   
   > anti:= 1/sqrt(2*tanh(x))+sqrt(2)/6*log( (1-sqrt(tanh(x)))/(1+sqrt(tanh(x))) )   
   >   
   > +sqrt(2)*arctan(tanh(x))+1/6 * arctan(sqrt(tanh(x)/2)):   
   >   
   > check:=simplify(integrand-diff(anti,x)); %too large to show   
   >   
   > plot(%,x=-Pi..Pi);   
   >   
   >   
   >   
   > --Nasser   
      
   A possible fix for #66 is:   
      
   restart:   
   integrand:=(cosh(x)^3 - sinh(x)^3) / (cosh(x)^3 + sinh(x)^3);   
   anti:= -1/(3*(1+tanh(x))) + 4/(3*sqrt(3))*arctan( (2*tanh(x)-1)/sqrt(3));   
   check:=simplify(integrand-diff(anti,x));   
      
   Did   
      
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