Evaluate _0^1 x(1-x)^5 d x - Vidyadip

Question

Evaluate $\int_0^1 x(1-x)^5 d x$

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Answer

$\text { Let } I =\int_0^1 x(1-x)^5 d x$
$=\int_0^1(1-x)[1-(1-x)]^5 d x \quad \ldots . .\left[\because \int_0^{ a } f (x) d x=\int_0^{ a } f ( a -x) d x\right]$
$=\int_0^1(1-x) x^5 d x$
$=\int_0^1\left(x^5-x^6\right) d x$
$=\int_0^1 x^5 d x-\int_0^1 x^6 d x$
$=\left[\frac{x^6}{6}\right]_0^1-\left[\frac{x^7}{7}\right]_0^1$
$=\frac{1}{6}\left(1^6-0\right)-\frac{1}{7}\left(1^7-0\right)$ $=\frac{1}{6}-\frac{1}{7}$
$\therefore I=\frac{1}{42}$

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