Question

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

Answer 1

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

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`