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Home  >>  CBSE XII  >>  Math  >>  Integrals
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By using the properties of definite integrals,evaluate the integral\[\int\limits_0^1x\;(1-x)^n\;dx\]

$\begin{array}{1 1} \frac{1}{(n+1)(n+2)} \\ \frac{1}{(n-1)(n-2)} \\ \frac{n+1}{n+2} \\ \frac{n}{(n+1)(n+2)} \end{array} $

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  • (i)$ \int \limits_a^b f(x)dx=F(b)-F(a)$
  • (ii)$\int \limits_0^a f(x)dx=\int \limits_0^a f(a-x)dx$
By property (ii)$\int \limits_0^a f(x)dx=\int \limits_0^a f(a-x)dx$
$\int\limits_0^1\;x(1-x)^n\;dx$ can be written as
On seperating the terms
On integrating we get,
Appllying the limits we get,
On simplifying $\frac{n+2-n-1}{(n+1)(n+2)}=\frac{1}{(n+1)(n+2)}$
answered Feb 13, 2013 by meena.p
edited Feb 6, 2014 by rvidyagovindarajan_1

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