How do you integrate this? #int_0^1(x^4(1-x)^4)/(1+x^2)dx#

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Answer 1 · 2018-03-17T15:51:06

#int_0^1(x^4(1-x)^4)/(1+x^2)dx=22/7-pi#

Let

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

Expand the numerator:

#I=int_0^1(x^8-4x^7+6x^6-4x^5+x^4)/(x^2+1)dx#

Apply long division:

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

Integrate directly:

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

Hence

#I=22/7-pi#