Let's first simplify x^4[1 - x]^4 / [1 + x^2]
[1 - x]^4 = [x - 1]^4 = x^4 - 4x^3 + 6x^2 - 4x + 1
So ..... x^4[ 1 - x]^4 / [ 1 + x^2] = x^4 [ x^4 - 4x^3 + 6x^2 - 4x + 1] / [1 + x^2] =
[ x^8 - 4x^7 + 6x^6 - 4x^5 + x^4] / [ 1 + x^2]
Perform synthetic division
x^6 - 4x^5 + 5x^4 - 4x^2 + 4
x^2 + 1 [ x^8- 4x^7 + 6x^6 - 4x^5 + x^4]
x^8 x^6
_________________________
-4x^7 + 5x^6 - 4x^5 + x^4
-4x^7 -4x^5
_______________________
5x^6 + x^4
5x^6 + 5x^4
__________________
-4x^4
-4x^4 - 4x^2
___________
4x^2
4x^2 + 4
_______
-4
So......we have
1
∫ x^6 - 4x^5 + 5x^4 - 4x^2 + 4 - 4 / [ x^2 + 1] dx =
0
1 1 1 1 1 1
x^7/7 ] - (2/3)x^6 ] + x^5 ] - (4/3)x^3 ] + 4x ] - 4arctan(x) ] =
0 0 0 0 0 0
(1/7) - (2/3) + 1 - (4/3) + 4 - 4arctan(1) =
22/7 - 4 [pi / 4] =
22/7 - pi ≈ 0.0013
