Question #54489

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Answer 1 · 2017-07-23T16:18:12

see below

Expand then divide.

Expand the numerator

#x^4(1-x)^4 = x^8-4x^7+6x^6-4x^5+x^4#

Do the division

#(x^4(1-x)^4)/(x^2+1) = (x^8-4x^7+6x^6-4x^5+x^4)/(x^2+1)#

# = x^6-4x^5+5x^4-4x^2+4 - 4/(x^2+1)#

Integrate:

#int_0^1 (x^4(1-x)^4)/(x^2+1) dx = int_0^1 ( x^6-4x^5+5x^4-4x^2+4 - 4/(x^2+1))dx#

# = [x^7/7-(2x^6)/3+x^5-(4x^3)/3+4x-4tan^-1x]_0^1#

Do the arithmetic.

# = (1/7-2/3+1-4/3+4-4(pi/4)) - (0)#

# = (3-14+21-28+84)/21 - pi#

# = (108-42)/21 = 66/21 = 22/7#