Evaluate : `int_0^1x(1-x)^5dx`

Explanation

`I=int_{0}^{1} x(1-x)^{5} d x`
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int_{a}^{b} f(x)=int_{a}^{b} f(a+b-x) d x`
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therefore I=int_{0}^{1}(1-x)(1-(1-x))^{5} d x`
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=int_{0}^{1}(1-x) x^{5} d x`
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=int_{0}^{1}(x^{5}-x^{6}) d x`
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=[frac{x^{6}}{6}-frac{x^{7}}{7}]_{0}^{1}=frac{1}{6}-frac{1}{7}=frac{1}{42}`
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