Suppose $f$ has at least two continuous derivatives, $f'$ is monotonically increasing, and $f' \geq \lambda$ for some $\lambda > 0$. How might one find the upper bound $|\int_a^b \cos(f(x))| \leq 2/\lambda$?
I've tried a number of basic approaches and none have worked. For instance, I've tried rewriting $\cos(f(x))$ as $(\sin(f(x)))' / f'(x)$, which seemed promising until I realized that I cannot justify the inequality $|\int_a^b (\sin(f(x)))' / f'(x)| \leq (1/\lambda) |\int_a^b (\sin(f(x)))'|$. Of course, were that true, then the result would follow quickly by evaluating the new integral, applying the triangle inequality, and finally noting that $|\sin(x)| \leq 1$.
I've tried a similar approach using the mean value theorem for integrals but was only able to show that there is some $\theta$, $a < \theta < b$, so that
$|\int_a^b \cos(f(x))| \leq \frac{1}{\lambda} (|f(\theta) - f(a)| + |f(b) - f(\theta)|)$,
which is not strong enough either.