Since $\int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1}$ for $k\neq -1$, replace $\int t\mathrm{d}t$ with $\frac{t^{2}}{2}$.
$$\frac{t^{2}}{2}$$
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.
