Cauchy-Schwarz Integral Inequality

Let $f(x)$ and $g(x)$ by any two Real integrable functions of $[a,b]$, then

$$\left[{\int_a^b f(x)g(x)\,dx}\right]^2\leq\left[{\int_a^b f^2(x)\,dx}\right]\left[{\int_a^b g^2(x)\,dx}\right],$$

with equality Iff $f(x)=kg(x)$ with $k$ real.

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1099, 1993.