Learning Fourier Integral

Question

I am learning Fourier Integral for real numbers. I downloaded a presentation of some university. I am summing up what I know and hopefully someone will correct me where I am wrong.

Fourier integral can be calculated only for functions that are decaying (what ever that means) and non-periodic

The formula for Fourier Integral : $$ f(x) = \int_{0}^{\infty}(A(\lambda)Cos(\lambda x)+B(\lambda)Sin(\lambda x)))\ d\lambda $$ where $\lambda$ is real numbers greater than or equal to zero.

Also my question is:
Fourier Series allows you to express a piece-wise continuous, periodic function as a sum of infinite terms. What does Fourier integral do ?

Answer 1

only for functions that are decaying (what ever that means) and non-periodic

Basically, yes. "Decaying" can be interpreted as "integrable" or "square integrable" (in terms of function spaces, $L^1$ or $L^2$). "Non-periodic" is redundant, since a function integrable in any sense cannot be periodic unless it's identically zero.

What does Fourier integral do ?

It helps us solve differential equations, by converting differentiation to multiplication (similarly to its close relative Laplace transform). It helps the folks over at Digital Signal Processing analyse sound, images, and other kinds of signals. Other ways to use the Fourier transform are listed at Wikipedia.