Difference between Fourier integral and Fourier transform 
What is the difference between Fourier integral and Fourier transform?

I know that for Fourier integral, the function must satisfy
$\int_{-\infty}^\infty  |f(t)| dt < \infty$, but what if I have a function that satisfies this condition: what does it mean to calculate Fourier transform and Fourier integral?


*

*Fourier Integral: 
$$f(t) =  \int_0^ \infty  A(\omega)\cos( \omega t) + B( \omega )\sin( \omega t)  d \omega$$
where
$A(\omega ) =  \frac{1}{\pi}   \int_{- \infty }^\infty f(t)\cos( \omega t) dt$ and $B(\omega ) = \frac{1}{\pi}  \int_{- \infty }^ \infty  f(t) \sin( \omega t) dt$

*Fourier Transform: 
$$F(\omega ) =  \frac{1}{2 \pi }   \int_{- \infty }^\infty  f(t) e^{-i \omega t}  dt$$

Your question is a bit ambiguous, since you don't state what you mean by fourier integral and fourier transform.
One possible source of confusion is that, while the fourier transform is indeed a linear isometry on $L^2$, the integral $$
  \int_{-\infty}^\infty f(t)e^{-i2\pi\omega t} \,dt
$$
does not converge for every $f \in L^2$. It does, however converge for every $f \in L^1 \cap L^2$, and the fourier transform on the full space $L^2$ can therefore be defined as the unique extension of the transform defined by the integral on $L^1 \cap L^2$. The result is then sometimes called the Fourier-Plancherel-Transform, but sometimes also simply the fourier transform on $L^2$.
Or you could simply be referring to the difference between the integral one uses to compute the coefficients of a fourier series and the integral used to define the fourier transform (on $L^2 \cap L^2$).
