NIMCET 2017 — Mathematics PYQ

Concept:  
${\int }_0^{a}f(x)dx={\int }_0^{a}f(a-x)dx$

Calculation:

Let, $I={\int }_0^{1}x(1-x{)}^{n}dx$

Using ${\int }_0^{a}f(x)dx={\int }_0^{a}f(a-x)dx$

$I={\int }_0^1(1-x)(1-(1-x){)}^{n}dx$

$I={\int }_0^1(1-x)x^{n}dx$

$I={\int }_0^1{x}^n-x^{n+1}dx$

$={\left[\frac{x^{n+1}}{n+1}-\frac{x^{n+2}}{n+2}\right]}_0^1$

$=\frac{1}{n+1}-\frac{1}{n+2}$

$=\frac{1}{(n+1)(n+2)}$

Hence, option (2) is correct.