29th Putnam 1968

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Problem A1

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