Answer
$$ - 410$$
Work Step by Step
$$\eqalign{
& \int_{ - 1}^1 {\left( {x - 1} \right){{\left( {{x^2} - 2x} \right)}^7}dx} \cr
& {\text{substitute }}u = {x^2} - 2x,{\text{ }}du = 2\left( {x - 1} \right)dx \cr
& {\text{express the limits in terms of }}u \cr
& x = - 1{\text{ implies }}u = {\left( { - 1} \right)^2} - 2\left( { - 1} \right) = 3 \cr
& x = 1{\text{ implies }}u = {\left( { - 1} \right)^2} - 2\left( { - 1} \right) = - 1 \cr
& {\text{the entire integration is carried out as follows}} \cr
& \int_{ - 1}^1 {\left( {x - 1} \right){{\left( {{x^2} - 2x} \right)}^7}dx} = \frac{1}{2}\int_3^{ - 1} {{u^7}du} \cr
& {\text{find the antiderivative}} \cr
& = \frac{1}{2}\left. {\left( {\frac{{{u^8}}}{8}} \right)} \right|_3^{ - 1} \cr
& {\text{use the fundamental theorem}} \cr
& = \frac{1}{{16}}\left( {{{\left( { - 1} \right)}^8} - {{\left( 3 \right)}^8}} \right) \cr
& {\text{simplify}} \cr
& = \frac{1}{{16}}\left( {1 - 6561} \right) \cr
& = - 410 \cr} $$
