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Home  >>  JEEMAIN and AIPMT  >>  Mathematics  >>  Class12  >>  Integral Calculus
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Integrate $\int \limits_0^1 x(1-x)^n dx. n \in N$

\[\begin {array} {1 1} (a)\;\frac{1}{n}-\frac{1}{n+1} \\ (b)\;\frac{1}{n+1}-\frac{1}{n+2} \\ (c)\;\frac{1}{n+2}-\frac{1}{n+3} \\ (d)\;None \end {array}\]

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1 Answer

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$\int\limits _0^1 x(1-x)^n dx$
=>$ 1-x=t=>-dx=dt$
at $x=0$. upper limit=1
at $x=1$. lower limit=0
=> $\int \limits _1^0 (1-t).f^n.(-dt)$
=> $- \int \limits_0^1 t^n.dt+ \int \limits _1^0 t^{n+1}.dt$
=> $ -\bigg(\large\frac{t^{n+1}}{n+1}\bigg)^0_1+\frac{t^{n+2}}{n+2}\bigg]_1^0$
=> $ \bigg(\large\frac{1}{n+1}$$+0-\large\frac{1}{n+2} \bigg)$
answered Dec 12, 2013 by meena.p
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