Question
using integration by partial fraction, prove that pi + integral from 0 to 1 of x^4(1-x)^4/1+x^2 = 22/7
Answers
Steve
Using long division, we know that
x^4(1-x)^4/(1+x^2) = x^6-4x^5+5x^4-4x^2+4 - 4/(x^2+1)
So, the integral is just
x^7/7 - 2x^6/3 + x^5 - 4x^3/3 + 4x - 4arctan(x)
evaluated at 1 and 0, we have
1/7 - 2/3 + 1 - 4/3 + 4 - π = 22/7 - π
now add π to that and you wind up with 22/7, or approximately π!
x^4(1-x)^4/(1+x^2) = x^6-4x^5+5x^4-4x^2+4 - 4/(x^2+1)
So, the integral is just
x^7/7 - 2x^6/3 + x^5 - 4x^3/3 + 4x - 4arctan(x)
evaluated at 1 and 0, we have
1/7 - 2/3 + 1 - 4/3 + 4 - π = 22/7 - π
now add π to that and you wind up with 22/7, or approximately π!