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(b) Let I, = sin2n+ (20) d0, n = 0, 1, ... . 0 2n (i) Prove that I, = 2n + 1 'n-1, n21. (ii) Deduce that I, = 2 2n ( n! ) 2 (2n + 1 )! Let J. = x" (1 - x)" dx, n = 0, 1, 2, ... . 0 ( n !) 2 (iii) Using the result of part (ii), or otherwise, show that J. = (2n + 1)! (iv) Prove that (2" n! ) s (2n + 1)