Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This is a homework excercise:

$f(x)$ is monotonic increasing and positive in $[a,\infty)$, $\lim_{a\rightarrow \infty}f(x)=\infty$, and $f'(x)$ is continuous in $[a,\infty)$. Prove $\int_{a}^{\infty}(f'(x)/f(x))\sin(f(x))dx$ converges.

I have solved this exercise, but I believe I have a mistake since I did not use the fact that $f'(x)$ is continuous. I would like to ask whether my solution is correct:

Solution (in brief)

By substituting $u=f(x)$ we get $\int_{a}^{\infty}(f'(x)/f(x))\sin(f(x))dx \rightarrow \int_{f(a)>0}^{f(\infty)=\infty}(\sin(u)/u)du$.

The integral $\int_{f(a)}^{N>\max(f(a),1)}(\sin(u)/u)du$ converges (due to continuity), and the integral $\int_{N}^{\infty}(\sin(u)/u)du$ converges by the Dirichlet test ($f(x)=1/x$ and $g(x)=\sin(x)$).

As you can see I didn't use the fact that $f'(r)$ is continuous anywhere. Is this solution correct?

share|cite|improve this question
up vote 1 down vote accepted

You've used $f'(x)$ continuous implicitly in using u-substitution. Per the wikipedia page for Integration by Substitution the function you're replacing has to be a continuously differentiable function. Continuously differentiable implies that the derivative is continuous as well as the function in question. As $f$ is supposed to meet those conditions, your proof is correct.

share|cite|improve this answer
Aah, damn, I knew I was being too optimistic with subsitution. – roel44 Dec 12 '11 at 0:09

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.