# Determine if $\int_2^\infty \frac{\sin x}{x\ln x}dx$ is convergent or not [duplicate]

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?

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

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.
• Just curious, why $100$? $2$ would do just as well.