Answer
$\dfrac{38229}{286}$
Work Step by Step
We will solve the given integral by using u-substitution method.
Let us consider that $u=x+1 \implies dx=du$
$\displaystyle \int_{0}^1 x^2 (x+1)^{10} \ dx= \int_0^1 (u-1)^2 u^{10} du$
or, $= \int_0^1 (u^{12}-2u^{11}+u^{10}) du$
or, $=[\dfrac{u^{13}}{13}-\dfrac{u^{12}}{6}+\dfrac{u^{11}}{11}+C]_0^1$
or, $=[\dfrac{(x+1)^{13}}{13}-\dfrac{(x+1)^{12}}{6}+\dfrac{(x+1)^{11}}{11}]_0^1$
or, $=[\dfrac{(1+1)^{13}}{13}-\dfrac{(1+1)^{12}}{6}+\dfrac{(1+1)^{11}}{11}]-[\dfrac{(0+1)^{13}}{13}-\dfrac{(0+1)^{12}}{6}+\dfrac{(0+1)^{11}}{11}]$
or, $=\dfrac{38229}{286}$
