29th Putnam 1968

Prove that ∫₀¹ x⁴(1 - x)⁴/(1 + x²) dx = 22/7 - π.

Solution

Divide the denominator by the numerator to get: (x⁸ - 4x⁷ + 6x⁶ - 4x⁵ + x⁴) = (1 + x²)(x⁶ - 4x⁵ + 5x⁴ - 4x² + 4) - 4.

Now we can integrate to get (x⁷/7 - 2x⁶/3 + x⁵ - 4x³/3 + 4x)|₀¹ - ∫₀¹ 4dx/(1 + x²) = 22/7 - π_.

29th Putnam 1968

© John Scholes
jscholes@kalva.demon.co.uk
14 Jan 2002