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Problem:

Determine if $\int_2^\infty \frac{\sin x}{x\ln x}dx$ converges or diverges.

What I tried:

I tried to IBP first, but it has $\sin x\ln(\ln x)$ so if $x\to \infty$, $\sin x\ln(\ln x)\to\infty$ so I thought $\int_2^\infty \frac{\sin x}{x\ln x}dx = \infty$ but WA and my book says that it converges.

Did I make a mistake? Or is my book wrong?

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    $\begingroup$ What is the full form of IBP ? $\endgroup$
    – Mantha Sai Gopal
    Sep 2, 2021 at 8:01
  • $\begingroup$ Have a look at mathsci.kaist.ac.kr/~kdryul/files/articles/… $\endgroup$
    – Gary
    Sep 2, 2021 at 8:20
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    $\begingroup$ How did you get $\sin x \ln \ln x \to \infty$? It is $0$ at integer multiples of $\pi$. $\endgroup$
    – Gary
    Sep 2, 2021 at 8:23
  • $\begingroup$ Why not compare it with $\int_2^{\infty} \frac{\sin x}{x} \ dx$? $\endgroup$
    – Toby Mak
    Sep 2, 2021 at 8:34
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    $\begingroup$ @TobyMak It could be problematic because we do not have absolute convergence for these integrals. $\endgroup$
    – Gary
    Sep 2, 2021 at 8:54

1 Answer 1

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The function $f(x) =\frac{1}{x\ln x}$ is monotonically decreasing and tends to $0$ at infinity. The function $g(x)=\sin x$ ha the property that $$\left|\int_2^{A} g(x)dx \right|\leq 100$$ for all $A>2.$ Hence the integral $$\int_2^{\infty} f(x) g(x) dx$$ converges by Dirichlet criterion.

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    $\begingroup$ Just curious, why $100$? $2$ would do just as well. $\endgroup$
    – Gary
    Sep 2, 2021 at 8:53

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