Factor out the constant using $\int af\left(y\right)\mathrm{d}y=a\int f\left(y\right)\mathrm{d}y$.
$$a^{2}\int y\mathrm{d}y$$
Since $\int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1}$ for $k\neq -1$, replace $\int y\mathrm{d}y$ with $\frac{y^{2}}{2}$.
$$a^{2}\times \frac{y^{2}}{2}$$
Simplify.
$$\frac{a^{2}y^{2}}{2}$$
If $F\left(y\right)$ is an antiderivative of $f\left(y\right)$, then the set of all antiderivatives of $f\left(y\right)$ is given by $F\left(y\right)+C$. Therefore, add the constant of integration $C\in \mathrm{R}$ to the result.
