How do you integrate this? int_0^1(x^4(1-x)^4)/(1+x^2)dx10x4(1x)41+x2dx

1 Answer
Mar 17, 2018

int_0^1(x^4(1-x)^4)/(1+x^2)dx=22/7-pi10x4(1x)41+x2dx=227π

Explanation:

Let

I=int_0^1(x^4(1-x)^4)/(1+x^2)dxI=10x4(1x)41+x2dx

Expand the numerator:

I=int_0^1(x^8-4x^7+6x^6-4x^5+x^4)/(x^2+1)dxI=10x84x7+6x64x5+x4x2+1dx

Apply long division:

I=int_0^1(x^6-4x^5+5x^4-4x^2-4/(x^2+1)+4)dxI=10(x64x5+5x44x24x2+1+4)dx

Integrate directly:

I=[1/7x^7-2/3x^6+x^5-4/3x^3-4tan^(-1)x+4x]_0^1I=[17x723x6+x543x34tan1x+4x]10

Hence

I=22/7-piI=227π