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Question:

$\int_{0}^{1} x(1-x)^{n} d x$

Solution:

Let $I=\int_{0}^{1} x(1-x)^{n} d x$

$\therefore I=\int_{0}^{1}(1-x)(1-(1-x))^{n} d x$

$=\int_{0}^{1}(1-x)(x)^{n} d x$

$=\int_{0}^{d}\left(x^{n}-x^{n+1}\right) d x$

$=\left[\frac{x^{n+1}}{n+1}-\frac{x^{n+2}}{n+2}\right]_{0}^{1}$    $\left(\int_{0}^{\infty} f(x) d x=\int_{0}^{0} f(a-x) d x\right)$

$=\left[\frac{1}{n+1}-\frac{1}{n+2}\right]$

$=\frac{(n+2)-(n+1)}{(n+1)(n+2)}$

$=\frac{1}{(n+1)(n+2)}$

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