Integrate $\int \limits_0^1 x(1-x)^n dx. n \in N$

Question

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

Answer 1

$\int\limits _0^1 x(1-x)^n dx$

=>$ 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)$