An integral inequality involving increasing function

Question

Let $0\leq a< b \leq \pi/2$

Let $f:[a,b]\to\mathbb R$ be a positive, increasing function.

Prove that $\left|\int_a^b f(t)\cos(t)dt\right|\leq f(b)(b+\sin(b))-f(a)(b+\sin(a))$

I haven't made any progress. I tried integration by parts, to no avail. I noted the inequality isn't symetric w.r.t $a$ and $b$.

Thanks for any hint.

Answer 1

If $f$ is increasing is almost everywhere differentiable. Since: $$\frac{d}{db}\left[f(b)(b+\sin(b))-f(a)(b+\sin a)\right]=(f(b)-f(a))+f(b)\cos b+(b+\sin b)\,f'(b)$$ and $f(b)-f(a)>0,b+\sin b>0,f'(b)\geq 0$ since $f$ is increasing, by integrating both sides of: $$ \frac{d}{db}\left[f(b)(b+\sin(b))-f(a)(b+\sin a)\right] > f(b)\cos b $$ we get our claim.