_0^1 x(1-x)^n d x= - Vidyadip

MCQ

$\int_0^1 x(1-x)^n d x=$

A
$\frac{n+3}{(n+1)(n+2)}$
B
$\frac{4}{(n+1)(n+2)}$
C
$\frac{2 n+3}{(n+1)(n+2)}$
D
$\frac{1}{(n+1)(n+2)}$

✓

Answer

$\begin{aligned} & \text {(d) : } \int_0^1 x(1-x)^n d x=\int_0^1(1-x)(1-(1-x))^n d x \\
& \left.\qquad \int_a^b f(x) d x=\int_a^b f(a+b-x) d x\right] \\
& =\int_0^1(1-x) x^n d x=\int_0^1\left(x^n-x^{n+1}\right) d x \\
& =\left[\frac{x^{n+1}}{n+1}-\frac{x^{n+2}}{n+2}\right]_0^1=\frac{1}{n+1}-\frac{1}{n+2} \\
& =\frac{n+2-n-1}{(n+1)(n+2)}=\frac{1}{(n+1)(n+2)}\end{aligned}$

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