Answer
$\approx 117.6426$
Work Step by Step
We will solve the given integral by using u-substitution method.
Let us consider that $u=x+1 \implies dx=du$
$\displaystyle \int_{0}^1 x^3 (x+1)^{10} \ dx= \int_0^1 (u-1)^3 u^{10} du$
or, $= \int_0^1 (u^{13}-3u^{12}-u^{10}) du$
or, $=[\dfrac{u^{14}}{14}-\dfrac{3u^{13}}{13}+\dfrac{u^{12}}{4}-\dfrac{u^{11}}{11}]_0^1$
or, $=[\dfrac{(x+1)^{14}}{14}-\dfrac{3(x+1)^{13}}{13}+\dfrac{(x+1)^{12}}{4}-\dfrac{(x+1)^{11}}{11}]_0^1$
or, $\approx 117.6426$
