Review integrals with sqrt.

Example (Template 4)

Int(1/(x^2sqrt(9x^2+1)))  (Test 1, Part I, 6)

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6.3 Partial Fractions (integrals of rational functions).
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Int((bm * x^m + ... +b1 * x + b0)/(an * x^n + ... + a1 * x + a0 ))

 The main condition m < n. "m" has to be less than "n". If not then 
use division of polynoomials.


Partial fractions are constructed out of the following four types of pieces.


  SWITCH(structure of the denominator){

   CASE I: splitted into distinct linear factors (p. 328)

         an * x^n + ... + a1 * x + a0 = (c1 * x + d1)* ...*(cn * x + dn)

?roots({148,24,1})={-12,2;-12,-2}

roots({148,24,1})
p(x) = 148.0 + 24.0*X^1 + 1.0*X^2
{-12,2}
{-12,-2}

   sum( Aj / (cj * x + dj), j=1...n)

V Example 2 on p. 329
Int((4*y^2-6*y-12)/(y*(y+2)*(y-3)),y={simpson,0.0001,1..2})
HW. 6.3, 1, 2, 12, 14

  CASE II:  splitted into linear factors some of which are repeated (p. 330)

  HW. 6.3, 3, 9, 15

1-(26*16+17*5)=-500/20=-25

  CASE |||: splitted into irreducible linear and quadratic factors, 
                  none of which is repeated. (p. 331)

 V Example. 4 on p. 331

   HW. 6.3, 4, 5, 6, 13, 16

   CASE IV: irreducible linear and quadratic factors some of which are repeated (p. 333)