How do you integrate this? int_0^1(x^4(1-x)^4)/(1+x^2)dx
1 Answer
Mar 17, 2018
Explanation:
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