Let $\varphi$ be a real smooth function with sufficient decay and vanishing mean. Then there exist a functions $\psi$ such that: $$\psi(x)'=\varphi(x)$$ then $$\hat{\psi}(\xi) = \frac{\hat{\varphi}(\xi)}{\xi} + \hat{\varphi}(0)\delta(\xi) = \frac{\hat{\varphi}(\xi)}{\xi}$$
Here i am using the following $$\hat\psi(\xi)=\int_{\mathbb{R}}\psi(x)e^{-2\pi i x\xi}\:dx$$
Can somehow help me understanding this result? Can someone give me a hint of how to prove this formula? Where does the delta function comes from?