Compute $\displaystyle\int\limits_{t-2T}^T t-\tau\cdot 1 \;d\tau = [t\tau – \frac{\tau^2}{2}]_{t-2T}^{T}$

Compute $ \displaystyle\int\limits_{t-2T}^T t-\tau\cdot 1 \;d\tau$

\begin{align} \displaystyle&\int\limits_{t-2T}^T t-\tau\cdot 1 \;d\tau \ &=\left[t\tau – \frac{\tau^2}{2}\right]_{t-2T}^{T}\ &=tT-\frac{T^2}{2}-t(t-2T)-\frac12(t-2T)^2\ &=tT-\frac{T^2}{2}-t^2+2tT-\frac 12(t^2-4tT+4T^2)\ &=tT-\frac{T^2}{2}-t^2+2tT-\frac 1 2t^2+2Tt-2T^2\ &=\frac{-3T^2}{2}+5tT-\frac{5tT^2}{2} \end{align}

Is this correct?