Integration, just as derivation, reveals a new approach to proving the inequalities. Let's take a detailed view on inequalities solved by or involving intgrals.
Theorem 1. If for all \(x\in[a,b]\) the following inequality holds \[f(x)\geq g(x),\] then for all \(x\in[a,b]\) we have \[\int^x_af(t)\,dt\geq\int^x_ag(t)\,dt.\]
I won't present a formal proof, but rather just a simple image to ilustrate the idea.
Now armed with this theorem let's solve few very basic problems.
Problem 1. Prove the following inequality
Solution. We know the very basic inequality for . Now if we substitute with we'll obtain . Now integrating our inequality we obtain Calculating we get
Problem 2. For prove the following inequality
Problem 3. For prove the following inequality