Finding the correct substitution for integral representaion

Question

I have as a part of a problem the following integrals

$\frac{1}{T}\int_{t-T}^t f(\tau) \, \mathrm{d}\tau = \frac{1}{T}\int_{0}^T f(t-u) \, \mathrm{d}u$, where $T > 0$ and $u \in [0,T]$

I cannot find the right substition. Can someone provide a hint?

Answer 1

We have $$ \int_{0}^{T}f(t-u)du \underset{\tau := t-u}{=} -\int_{t}^{t-T}f(\tau)d\tau = \int_{t-T}^{t}f(\tau)d\tau. $$