Answer
$$\frac{21}{4}$$
Work Step by Step
Given $$\int_0^1y(1+y^2)^5dy$$
Let $u=1+y^2\ \Rightarrow \ \ du=2ydy$ at $ y=0\to u=1$,at $ y=1\to u=2$
\begin{align*}
\int_0^1y(1+y^2)^5dy&=\frac{1}{2}\int_1^2u^5du\\
&=\frac{1}{12}u^6\bigg|_{1}^{2}\\
&=\frac{63}{12}=\frac{21}{4}
\end{align*}
