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Question
By using the properties of the definite integral, evaluate the integral:
`int_0^1 x(1-x)^n dx`
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Solution
`int_0^1 (1 - x) [1 - (1 - x)^n] dx ...[because int_0^a f(x) dx = int_0^a f(a - x) dx]`
Hence, `I = int_0^1 (1 - x).x^n dx`
`I = int_0^1 (x^n - x^(n + 1)) dx`
`= ([x^(n + 1)]_0^1)/(n + 1) - ([n^(n + 2)]_0^1)/(n + 2)`
`= 1/(n + 2) - 1/(n + 2)`
`= (n + 2 - n - 1)/((n + 1)(n + 2))`
`= 1/((n + 1)(n + 2))`
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