If we have two functions $f,g:[a,b]\to\mathbb{R}$ and we know they are bounded, so:
$\sup_{x\in[a,b]}|f(x)|=K$, and $\sup_{x\in[a,b]}|g(x)|=M$.
Where $K,M$, are positive finite constants, which of the following inequalities is the correct one?
$(1)$ $\int_a^b|f(x)||g(x)|dx\leq \sup_{x\in[a,b]}|f(x)|\sup_{x\in[a,b]}|g(x)|=KM$, or
(2) $\int_a^b|f(x)||g(x)|dx\leq \sup_{x\in[a,b]}|f(x)|\sup_{x\in[a,b]}|g(x)|\int_a^b1\,dx=KM(b-a).$
Thanks!