How do you integrate this? int_0^1(x^4(1-x)^4)/(1+x^2)dx∫10x4(1−x)41+x2dx
1 Answer
Mar 17, 2018
Explanation:
Let
I=int_0^1(x^4(1-x)^4)/(1+x^2)dxI=∫10x4(1−x)41+x2dx
Expand the numerator:
I=int_0^1(x^8-4x^7+6x^6-4x^5+x^4)/(x^2+1)dxI=∫10x8−4x7+6x6−4x5+x4x2+1dx
Apply long division:
I=int_0^1(x^6-4x^5+5x^4-4x^2-4/(x^2+1)+4)dxI=∫10(x6−4x5+5x4−4x2−4x2+1+4)dx
Integrate directly:
I=[1/7x^7-2/3x^6+x^5-4/3x^3-4tan^(-1)x+4x]_0^1I=[17x7−23x6+x5−43x3−4tan−1x+4x]10
Hence
I=22/7-piI=227−π