I came across an algebraic manipulation which I don't understand. Maybe someone can explain it to me.
For a fixed $x\in\mathbb{R}$ we are given the integral: $$\frac{1}{\pi}\int\limits_{-\pi}^{\pi}f(x+u)D_n(u)du,$$ where $f:\mathbb{R}\to\mathbb{R}$ is a $2\pi$-periodic function which is Riemann-integrable over $[-\pi,\pi]$. $D_n(u)$ is the $n$-th Dirichlet-kernel.
Our professor did the following manipulations where he substituted $u=-v$:
$$\frac{1}{\pi}\int\limits_{-\pi}^{\pi}f(x+u)D_n(u)du\underset{\text{substitution}}{=}\frac{-1}{\pi} \int\limits_{\pi}^{-\pi}f(x-v)D_n(-v)dv =\frac{1}{\pi} \int\limits_{-\pi}^{\pi}f(x-v)D_n(v)dv.$$
Where does the $-1$ in the fraction after the first equal sign come from?
When I did the manipulation I got:
$$\frac{1}{\pi}\int\limits_{-\pi}^{\pi}f(x+u)D_n(u)du\underset{\text{substitution}}{=}\frac{1}{\pi} \int\limits_{\pi}^{-\pi}f(x-v)D_n(-v)dv =\frac{-1}{\pi} \int\limits_{-\pi}^{\pi}f(x-v)D_n(v)dv.$$