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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?

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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.

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Aah, damn, I knew I was being too optimistic with subsitution. – roel44 Dec 12 '11 at 0:09

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