JEE Advanced · 2010 · 11 Apr 2010 · Paper 1 · Definite Integrals

Expand the numerator: $x^4(1-x{)}^4=x^4(1-4x+6x^2-4x^3+x^4)$. Perform polynomial long division of this by $1+x^2$ to separate the integrand into a polynomial part plus a remainder over $1+x^2$. The polynomial part integrates directly; the remainder $\frac{-4x^2}{1+x^2}$ can be written as $-4+\frac{4}{1+x^2}$. Integrating from 0 to 1: the polynomial terms yield $\frac{22}{7}$ and the arctangent term contributes $-\pi$. Hence the answer is $\frac{22}{7}-\pi$.