What is a "formal integral" I'm reading in my text on Fourier integrals and after presenting an integral in the form of a solution it says the following

of course this is a formal integral to be interpreted as a generalized
  function.

well, my response to the author is "not of course". What does the author mean by this statement? I feel like it's important I understand what they're saying in terms of interpreting the solution. I'm just looking for clarity, but if it helps to explain by example that may help as well.
 A: Without more context, my inclination is to believe the author means that the object is a “formal integral” in the same sense as a “formal power series,” or in this case maybe a formal Laurent series would be more if not fully appropriate.
In this case, questions of convergence are entirely irrelevant, as only the formal algebraic properties of the construction are of interest. The variables involved are no longer necessarily to be construed as “numbers,” for example, but instead are taken to be “formal indeterminates” that satisfy certain formal algebraic properties.
To comprehend this in a broader context, it may help to look into generating functions. A good book on that subject is “Generatingfunctionology” by Herbert Wilf.
Hopefully that’s helpful!
Connor
A: In Fourier analysis, the Dirac delta distribution $\delta(x)$ is a not formal at all thing essentially defined by $\int_{-\infty}^\infty \delta(x) \varphi(x)dx=\lim_{\epsilon \to 0} \int_{-\infty}^\infty \frac{1_{|x| < \epsilon}}{2 \epsilon}\varphi(x)dx $ which converges to $\varphi(0)$ whenever $\varphi$ is continuous. 
What is complicated is to know how the limit (if it exists) depends on the precise chosen sequence. The Fourier inversion theorem is that replacing $\frac{1_{|x| < \epsilon}}{2 \epsilon}$ by $\int_{-1/\epsilon}^{1/\epsilon} e^{2i \pi \xi x} d \xi$ works at least when $\varphi \in L^1, \hat{\varphi} \in L^1$.
Distribution theory/generalized functions is a way to generalize this to many other things than $\delta(x)$, assuming $\varphi \in C^\infty_c$.
