Windowed Fourier Transform I need some help with applying the properties of Fourier Transform. 

We define the windowed Fourier Transform of $f \in L^2(R)$ as 
$$Sf(\mu,\xi)=\int_\mathbb{R}f(t)g(t-\mu)e^{-i\xi t}dt$$

Prove that for $f=e^{i\eta_0t}$, we have $Sf(\mu,\eta)=e^{-i\mu(\eta-\eta_0)}\hat{g}(\eta-\eta_0)$ 

All I can obtain till now is this. 
$$Sf(\mu,\eta)=\int_\mathbb{R}e^{i\eta_0t}g(t-\mu)e^{-i\eta t}dt=\int_\mathbb{R}e^{-it(\eta-\eta_0)}g(t-\mu)dt$$

I know that I need to apply some properties of fourier transform but I am not sure how. Thanks in advance.
HINT:

Just substitute $\tau=t-\mu$ in the integral and see what you get. It's just very basic math, no complicated properties of the Fourier transform. I assume that you know the definition of $\hat{g}(\eta)$.