Examples for Section 7.3
Partial Fractions

p. 576

1.

Apart[((5  x  -  13)/((x - 3) (x - 2)))]

3.

Apart[((x + 4)/(((x + 1))^2))]

4.

Write (x^2) -  2  x  +  1=(((x - 1))^2).

Apart[((2  x + 2)/((x^2) - 2  x + 1))]

14.

(y^2) +  y  =  y(y + 1)

Apart[((y + 4)/((y^2) +  y))]

∫((4/y) - (3/(1 + y))) \[DifferentialD]y

(∫_(1/2)^1)((4/y) - (3/(1 + y))) \[DifferentialD]y

19.

\(TraditionalForm
    \`\((x\^2 - \ 1)\)\^2 = \ \(\((x - 1)\)\^2\) \((x + 1)\)\^2\)

Apart[(1/((((x^2) - 1))^2))]

∫(
          (1/(4 (((-1 + x))^2))) - (1/(4 (-1 + x))) + 
            (1/(4 (((1 + x))^2))) + (1/(4 (1 + x)))) 
        \[DifferentialD]x

Simplify[%]

∫(1/((((x^2) - 1))^2)) \[DifferentialD]x

24.

Apart[((8  (x^2) +  8  x  +  2)/(((4  (x^2) + 1))^2))]

∫(((8 x)/(((1 + 4 (x^2)))^2)) + (2/(1 + 4 (x^2)))) 
        \[DifferentialD]x

∫((8  (x^2) +  8  x  +  2)/(((4  (x^2) + 1))^2)) 
        \[DifferentialD]x

32.

4  (x^2) - 4  x + 1 = (((2  x - 1))^2)

Apart[((16  (x^3))/(4  (x^2) -  4  x + 1))]

∫(
          4 + 4 x + (2/(((-1 + 2 x))^2)) + (6/(-1 + 2 x))) 
        \[DifferentialD]x

∫((16  (x^3))/(4  (x^2) -  4  x + 1)) 
        \[DifferentialD]x

52.

Find a second degree polynomial P(x)  =  a (x^2)  +  b x  +  c such that P(0)  =  1, P' (0)  =  0, and \(TraditionalForm
      \`∫\(\(P(x)\)\/\(\(x\^3\)(x - 1)\)\^2\) 
          dx\) is a rational function.

Solution. 1 = P(0) = a (0^2)  +  b 0 + c, so c  =  1, and 0  =  P' (0)  =  2  a (0^2)  +  b, so b  =  0. Only a is undetermined at this point. Now \(TraditionalForm
      \`\(P(x)\)\/\(\(x\^3\)(x - 1)\)\^2 = \ 
        \(a\ x\^2 + 1\)\/\(\(x\^3\)(x - 1)\)\^2\)

Apart[((a (x^2) + 1)/((x^3) (((x - 1))^2)))]

∫(
          ((1 + a)/(((-1 + x))^2)) + ((-3 - a)/(-1 + x)) + 
            (1/(x^3)) + (2/(x^2)) + ((3 + a)/x)) \[DifferentialD]x

The only way to make this a rational function (i.e., a quotient of polynomials) is to get rid of the logarithms, and that can be done if a  =  -3. So the asnwer is P(x)  =  -3 (x^2)  +  1.