Ostrowski's Inequality

If $f(x)$ is a monotonically increasing integrable function on $[a,b]$ with $f(b)\leq 0$, then if $g$ is a Real function integrable on $[a,b]$,

$$\left\vert{\int_a^b f(x)g(x)\,dx}\right\vert\leq \vert f(a)\... ...ax_{a\leq\xi\leq b}\left\vert{\int_a^\xi g(x)\,dx}\right\vert.$$

References

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, p. 1100, 1979.