Problem 1 For each natural number n, let (1-x')" cos(Ox) dx. . Prove that 02In = 2n(2n- 1)1,-1 -4m(n -1)17-2 for all n 2 2, and hence 02"+I, = n!(P,, (@)sin(0) + Q,(0) cos(0)), where P, and Q, are polynomials of degree at most 2n with integer coefficients. . Deduce that i is irrational