Let f and h be two integrable functions in [a,b]. Let M=sup|f(x)| when x in [a,b].
Prove that:
| int{f(x)*h(x)}dx | <=M*int{| h(x) |}dx
(The both integrals are from a to b)
We have $\displaystyle \left|\int_a^bf\left(x\right)g\left(x\right)dx \right|\leq \int_a^b\left|f\left(x\right)g\left(x\right)\right | dx =\int_a^b\left|f\left(x\right)\right|\left|g\left( x\right)\right| dx \leq \int_a^bM\cdot \left|g\left(x\right)\right| dx $