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Let rk = 01 (1 - x7 )k dx 01 (1 - x7 )k+1 dx, k N. Then the value of k=110 17 (rk - 1 ) is equal to .
65
Let Ik = 01 (1 - x7 )k dx, so rk = IkIk+1
By parts (taking 1 as the second function):
Ik+1 = [x (1 - x7 )k+1 ]01 + 7(k + 1) 01 x7 (1 - x7 )k dx
x7 = 1 - (1 - x7 ): Ik+1 = 7(k + 1) (Ik - Ik+1 )
rk = 7k + 87k + 7 rk - 1 = 17(k + 1) 17 (rk - 1 ) = k + 1
k=110(k + 1) = 2 + 3 + + 11 = 65
Answer: 65
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