First of all, it certainly isn't true that $f$ is Lebesgue integrable implies $f$ is bounded. Thus I will assume by integrable we mean Riemann Integrable.
With the definition of the Riemann Integral given in, for example, Rudin's Principles of mathematical Analysis and Abbott's Understanding Analysis, a proof of this fact is unnecessary as both require that $f$ is bounded for in order for it to even enter in the definition for the Riemann integral.
In these books, the integral is only defined for bounded functions so if we have a function and we know how to integrate it, then it has to be bounded.
For references, see Definition 6.2 on page 122 of Rudin or Section 7.2 starting on page 186 in Abbot's text (page numbers, etc may differ by edition).
Edit: It has been brought to my attention by @SantiagoCanez in the comments that in Real Mathematical Analysis, Pugh defines the Riemann Integral in a way that does not definitionally require boundedness. The first theorem Pugh proves once he defines the Riemann Integral is that integrability implies boundedness.
This is Theorem 15 on page 155 in my edition. This goes to show that one must first agree on definitions. I don't know how you define the Riemann Integral, but see here for Pugh's proof that integrability implies boundedness (and scroll up for his definition of the Riemann Integral.)
Considering this, and the discussion below, certain definitions require a proof that integrability implies boundedness. For one such proof, see @ReneSchipperus's answer.