p.578, 51.

51.

a) Evaluate \(TraditionalForm
      \`∫\_0\%1\(\(\( x\^4\)(x - 1)\)\^4\/\(x\^2 + 1\)\) 
          dx\)

Expand[(x^4) (((x - 1))^4)]

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

∫(4 - 4 (x^2) + 5 (x^4) - 4 (x^5) + (x^6) - (4/(1 + (x^2))))
           \[DifferentialD]x

(∫_0^1)
        (4 - 4 (x^2) + 5 (x^4) - 4 (x^5) + (x^6) - (4/(1 + (x^2)))) 
        \[DifferentialD]x

(∫_0^1)(( (x^4) (((x - 1))^4))/((x^2) + 1)) 
        \[DifferentialD]x

b)

N[(22/7) - π]

Pecentage of π:

100  N[(((22/7) - π)/π)]

c)

p = Plot[(((x^4) (((x - 1))^4))/((x^2) + 1)), {x, 0, 1}]

The maximum value appears to be around 0.003, a very small number. This indicates that the total area under this curve, (22/7) - π, is also very small. With a more "normal" scale on the y-axis, the graph would be difficult to see.

Show[p,  PlotRange -> {0, 0.5}]

Magnified 10 times:

Show[p,  PlotRange -> {0, 0.1}]

Magnified 100 times:

Show[p,  PlotRange -> {0, 0.01}]