Question

The value of $\underset{2}{\overset{10}{\int }}(x-1)(x-2)(x-3)\cdots (x-11)dx$ is

Answer 1

$I=\underset{2}{\overset{10}{\int }}(x-1)(x-2)(x-3)(x-4)(x-5)\cdots (x-11)dx$
Put $x-6=t$
$\Rightarrow dx=dt\Rightarrow I=\underset{-4}{\overset{4}{\int }}(t+5)(t+4)(t+3)(t+2)(t+1)\cdots (t-5)dt\Rightarrow I=\underset{-4}{\overset{4}{\int }}t\cdot (t^2-25)(t^2-16)(t^2-9)(t^2-4)(t^2-1)dt\therefore I=0[\because \underset{-a}{\overset{a}{\int }}f\left(x\right)dx=0,\text{if}f\left(x\right)\text{is odd}]$