Originally Posted by
Laurent It seems that you're not expected to use a) to evaluate the integral. The simplest way anyway is probably to expand the numerator, and perform the Euclidian division of $\displaystyle X^4(1-X)^4$ by $\displaystyle 1+X^2$ (like with integers, you know?), and get something like $\displaystyle X^4(1-X)^4=Q(X)(1+X^2)+aX+b$ hence your integral equals $\displaystyle \int_0^1 Q(x)dx + a\int_0^1 \frac{x}{1+x^2}dx + b\int_0^1\frac{dx}{1+x^2}$. The first term is easy, the second term has antiderivative $\displaystyle \frac{1}{2}\log(1+x^2)$, and the third one relates to arctan.
For the second part, you probably have to write $\displaystyle 1\leq 1+x^2\leq 2$ hence $\displaystyle \frac{1}{2}x^4(1-x)^4\leq \frac{x^4(1-x)^4}{1+x^2}\leq x^4(1-x)^4$ and integrate.