Integration of Rational function.

Question

I have this problem $$ \int_{0}^{1} \frac{x^4(1-x)^4}{1+x^2}dx$$

This is problem from IITJEE 2000 Paper 1

I tried following steps

I tried using few properties but I know only few of them hence no luck there.

$\tan\theta = x $ but still stuck.

Integration by Parts still got nothing.

And Partial fractions. Still no help. Can anyone help me out. And also share the methodology for solving similar problems. Thanks.

Answer 1

Partial fractions should work. Long division gives that $$ \frac{x^4(1-x)^4}{1+x^2}=x^6-4x^5+5x^4-4x^2-\frac{4}{1+x^2}. $$ Integration of the polynomial portion should be straight forward and use the fact that $$ \frac{d}{dx}(\arctan x)=\frac{1}{1+x^2} $$ for the rest.

Answer 2

$$\frac{x^4(1-x)^4}{1+x^2}=Q(x) +\frac {a + bx}{1+x^2} .$$

Where $Q(x) $ is a polynomial.

The $$ \frac {a + bx}{1+x^2}$$ is also integrated easily.