I am evaluating the fourier transform of a function + a constant: $\frac{1}{2\pi}\int_{-\infty}^{+\infty}(f(x)+c)e^{-ikx}dx$ equals what?

Question

I am evaluating the fourier transform of a function plus a constant $c$:

$$\frac{1}{2\pi}\int_{-\infty}^{+\infty}(f(x)+c)e^{-ikx}dx.$$

As a result, I should get the fourier transform of the function $f(x)$ plus a dirac delta: $$\frac{1}{2\pi}\int_{-\infty}^{+\infty}(f(x)+c)e^{-ikx}dx=\hat{f}(k)+c\delta(k).$$

However, I cannot understand whether I am making some mistake or not and the meaning of such a result.