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ExampleIntegration2

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Editor: test1 Time: 2024/09/02 16:09:58 GMT+0
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Here we show how FriCAS can help solving calculus problems.
Task is: find integer n, $60 < n < 100$ such that x^n/(x^120 - 1)
has simple integral.

Theory of integration of rational functions says that integral
is sum of constant multiples of logarithms of factors of x^120 - 1.  We write
x^120 - 1 as product of four factor:
\begin{axiom}
)set output tex off
)set output algebra on
(x^30 + 1)*(x^30 - 1)*(x^30 + %i)*(x^30 - %i)
\end{axiom}

Parametric form of integral corresopnding to those factors is
and its derivative are:
\begin{axiom}
ii := a*log(x^30 - 1) + b*log(x^30 + 1) + c*log(x^30 + %i) + d*log(x^30 -%i)
f := D(ii, x)
\end{axiom}

We see that f is a linear combination of powers of x divided by x^120 - 1.  The
only power in rage is x^89.  So we need to find a, b, c, d such that
coefficients of other powers are zero.  To do this we extract coefficients
and solve relevant system of equation.  As a first step is it convenient
to convert f to a rational function with coefficient being complx polynomials,
then the other steps are easy:
\begin{axiom}
-- convert to rational function
rf := f::FRAC(POLY(COMPLEX(INT)))
-- extract numerator and coefficients of powers of x
nf := numer(rf)
lc := coefficients(univariate(nf, x))

-- solve systme of linear equations
sl := solve([lc(1), lc(2) - 1, lc(3), lc(4)], [a, b, c, d])
\end{axiom}

Finally we check obtaine result:
\begin{axiom}
ii1 := eval(ii, sl(1))
D(ii1, x)
\end{axiom}

Problem solved.

Here we show how FriCAS? can help solving calculus problems. Task is: find integer n, such that x^n/(x^120 - 1) has simple integral.

Theory of integration of rational functions says that integral is sum of constant multiples of logarithms of factors of x^120 - 1. We write x^120 - 1 as product of four factor:

fricas

(1) -> )set output tex off

fricas

)set output algebra on

(x^30 + 1)*(x^30 - 1)*(x^30 + %i)*(x^30 - %i)

         120
   (1)  x    - 1

Type: Polynomial(Complex(Integer))

Parametric form of integral corresopnding to those factors is and its derivative are:

fricas

ii := a*log(x^30 - 1) + b*log(x^30 + 1) + c*log(x^30 + %i) + d*log(x^30 -%i)

               30               30                30                30
   (2)  b log(x   + 1) + c log(x   + %i) + d log(x   - %i) + a log(x   - 1)

Type: Expression(Complex(Integer))

fricas

f := D(ii, x)

   (3)
                                   119                                     89
       (30 d + 30 c + 30 b + 30 a)x    + (30 %i d - 30 %i c - 30 b + 30 a)x
     + 
                                   59                                       29
     (- 30 d - 30 c + 30 b + 30 a)x   + (- 30 %i d + 30 %i c - 30 b + 30 a)x
  /
      120
     x    - 1

Type: Expression(Complex(Integer))

We see that f is a linear combination of powers of x divided by x^120 - 1. The only power in rage is x^89. So we need to find a, b, c, d such that coefficients of other powers are zero. To do this we extract coefficients and solve relevant system of equation. As a first step is it convenient to convert f to a rational function with coefficient being complx polynomials, then the other steps are easy:

fricas

-- convert to rational function
rf := f::FRAC(POLY(COMPLEX(INT)))

   (4)
                                   119                                     89
       (30 d + 30 c + 30 b + 30 a)x    + (30 %i d - 30 %i c - 30 b + 30 a)x
     + 
                                   59                                       29
     (- 30 d - 30 c + 30 b + 30 a)x   + (- 30 %i d + 30 %i c - 30 b + 30 a)x
  /
      120
     x    - 1

Type: Fraction(Polynomial(Complex(Integer)))

fricas

-- extract numerator and coefficients of powers of x
nf := numer(rf)

   (5)
                                 119                                     89
     (30 d + 30 c + 30 b + 30 a)x    + (30 %i d - 30 %i c - 30 b + 30 a)x
   + 
                                   59                                       29
     (- 30 d - 30 c + 30 b + 30 a)x   + (- 30 %i d + 30 %i c - 30 b + 30 a)x

Type: Polynomial(Complex(Integer))

fricas

lc := coefficients(univariate(nf, x))

   (6)
   [30 d + 30 c + 30 b + 30 a, 30 %i d - 30 %i c - 30 b + 30 a,
    - 30 d - 30 c + 30 b + 30 a, - 30 %i d + 30 %i c - 30 b + 30 a]

Type: List(Polynomial(Complex(Integer)))

fricas

-- solve systme of linear equations
sl := solve([lc(1), lc(2) - 1, lc(3), lc(4)], [a, b, c, d])

               1          1        %i         %i
   (7)  [[a = ---, b = - ---, c = ---, d = - ---]]
              120        120      120        120

Type: List(List(Equation(Fraction(Polynomial(Complex(Integer))))))

Finally we check obtaine result:

fricas

ii1 := eval(ii, sl(1))

               30                30                 30              30
        - log(x   + 1) + %i log(x   + %i) - %i log(x   - %i) + log(x   - 1)
   (8)  -------------------------------------------------------------------
                                        120

Type: Expression(Complex(Integer))

fricas

D(ii1, x)

            89
           x
   (9)  --------
         120
        x    - 1

Type: Expression(Complex(Integer))

Problem solved.