Answer
$-\dfrac{4}{21}$
Work Step by Step
We are given the integral $I=\int_1^2 x(x-2)^5 \ dx$
We will solve the given integral by using u-substitution method.
Let us consider that $u=x-2 \implies dx=du$
So, we can write as: $\int_1^2 x(x-2)^5 \ dx=\int_1^2 (u+2)u^5 \ du\\ = \int_1^2 (u^6+2u^5) \ du$
Now, use the following formula such as:
$\int x^n \ dx=\dfrac{x^{n+1}}{n+1}+C$
Therefore, we have:$\int_1^2 (u^6+2u^5) \ du=[\dfrac{u^7}{7}+\dfrac{u^6}{3}]_1^2+C$
or, $=[\dfrac{(x-2)^7}{7}+\dfrac{(x-2)^6}{3}]_1^2+C$
or, $=[\dfrac{(2-2)^7}{7}+\dfrac{(2-2)^6}{3}]-[\dfrac{(1-2)^7}{7}+\dfrac{(1-2)^6}{3}]$
or, $=-\dfrac{4}{21}$
