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Question: problem with integration,which one is correct !?

Question:problem with integration,which one is correct !?

mehdi jafari 764 Maple
integral
August 13 2016
0

hi every one , i have a problem with integration,which one is correct !? also when i remove epsilon option, it returns unevaluated integral . can i have crorrect and reliable answer?

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restart
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with(LinearAlgebra):
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Digits := 30;

30

 (1)

``

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N := 4;

4

  

proc (x, s) options operator, arrow; exp(x*s) end proc

  

proc (x) options operator, arrow; exp(2*x)+(exp(x*(x+2))-exp(-x-2))/(x+2) end proc

 (2)
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alpha := -1/2;

-1/2

  

-1/2

 (3)
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for n from 0 to N+1 do J[n] := unapply(simplify((-1)^n*(1-x)^(-alpha)*(1+x)^(-beta)*(diff((1-x)^(n+alpha)*(1+x)^(n+beta), [`$`(x, n)]))/(2^n*factorial(n))), x) end do;

proc (x) options operator, arrow; 1 end proc

  

proc (x) options operator, arrow; (1/2)*x end proc

  

proc (x) options operator, arrow; (3/4)*x^2-3/8 end proc

  

proc (x) options operator, arrow; (5/4)*x^3-(15/16)*x end proc

  

proc (x) options operator, arrow; (35/16)*x^4-(35/16)*x^2+35/128 end proc

  

proc (x) options operator, arrow; (315/256)*x+(63/16)*x^5-(315/64)*x^3 end proc

 (4)
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u := unapply(exp(2*x), x);

proc (x) options operator, arrow; exp(2*x) end proc

 (5)
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for i from 0 to N do phi[i] := J[i](x) end do

1

  

(1/2)*x

  

(3/4)*x^2-3/8

  

(5/4)*x^3-(15/16)*x

  

(35/16)*x^4-(35/16)*x^2+35/128

 (6)
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w1 := (1-x)^alpha*(1+x)^beta;

1/((1-x)^(1/2)*(1+x)^(1/2))

 (7)
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for j from 0 to N do S[j] := simplify(evalf(int(k(x, s)*subs(x = s, phi[j]), s = -1 .. x))) end do;

(-1.*exp(-1.*x)+1.*exp(x^2))/x

  

.5*(exp(x^2)*x^2+exp(-1.*x)*x+exp(-1.*x)-exp(x^2))/x^2

  

((.75*x^4-1.875*x^2+1.5)*exp(x^2)-.375*(x+2.)^2*exp(-1.*x))/x^3

  

((1.25*x^6-4.6875*x^4+8.4375*x^2-7.5)*exp(x^2)+(.3125*x^3+2.8125*x^2+7.5*x+7.5)*exp(-1.*x))/x^4

  

((2.1875*x^8-10.9375*x^6+30.8984375*x^4-56.875*x^2+52.5)*exp(x^2)+(-.2734375*x^4-4.375*x^3-21.875*x^2-52.5*x-52.5)*exp(-1.*x))/x^5

 (8)
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for i from 0 to N do evalf(Int(phi[i]*S[3]*w1, x = -1 .. 1, epsilon = 0.1e-24)) end do

-0.940736801761282474970573535984e-2

  

-0.499428361530268194608821964968e-1

  

-0.716662760275484038621082776589e-1

  

-0.175968374179810236471161325565e-3

  

0.205695155661022012897834537769e-1

 (9)
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for i from 0 to N do evalf(Student:-Calculus1:-ApproximateInt(phi[i]*S[3]*w1, x = -1 .. 1)) end do

0.112342629325121433520368350257e-1

  

-0.372785154811933200270010346236e-1

  

-0.625152339329758587804224387138e-1

  

0.880883783666879589242935631796e-2

  

0.300672960958837351790545896762e-1

 (10)
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NULL

 

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