Factor out the constant using $\int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x$.
$$\beta y\int x\mathrm{d}x$$
Since $\int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1}$ for $k\neq -1$, replace $\int x\mathrm{d}x$ with $\frac{x^{2}}{2}$.
$$\beta y\times \frac{x^{2}}{2}$$
Simplify.
$$\frac{\beta yx^{2}}{2}$$
The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.
