Dirichlet's Kernel in Fourier Analysis

Prove that 1/2$\displaystyle \pi$$\displaystyle \int$f(t)Dn(x-t)dt = 1/2$\displaystyle \pi$$\displaystyle \int$f(x-t)Dn(t)dt

the integral ranges from -$\displaystyle \pi$ to $\displaystyle \pi$

Dn represents the nth Dirichlet's Kernel, $\displaystyle \sum$ e^ikx as k ranges from -n to n

i tried a substitution where u = x - t which yield -du = dt

so the integral i am getting is

1/2$\displaystyle \pi$$\displaystyle \int$f(t)Dn(x-t)dt = - 1/2$\displaystyle \pi$$\displaystyle \int$f(x-u)Dn(u)du

any help would be welcomed