Differentiation in Schwartz Space, integration by parts

Question

I have a simple question about integration by parts in multi differentiation case.
Suppose $f\in S$ (Schwartz space, i.e. space of rapidly decreasing functions).
When proving the Fourier Transform:
$$\widehat{D^\alpha f}=\xi ^\alpha \hat f(\xi)$$ I use the notation $D_j=\frac{1}{i}\partial_j \text{ and } D^\alpha=D^\alpha_1 D^\alpha_2... $ and $\alpha$ is a multi-index
So we have
$$\widehat{D^\alpha f}(\xi)=\int e^{-ix\xi}D_x^\alpha f(x)dx=(-1)^{|\alpha|}\int D_x^\alpha e^{-ix\xi}f(x)dx=(-1)^{|\alpha|}\int (-1)^{|\alpha|}\xi^\alpha e^{-ix\xi}f(x)dx=\xi^\alpha \hat f(\xi)$$ I guess that the second equal sign comes from integration by parts, but could someone explain more in details? I mean there should be a constant part (without the integral sign) in the integration by part formula?

And I'm also confused about the third equal sign, could someone help me understand?

I greatly appreciate your time and help!