# Explore absolute and conditional convergence of integral $\int_{1}^{+\infty} \frac{\sin \sqrt x}{x-\ln x}\, dx$

Explore absolute and conditional convergence of the integral $$\int_{1}^{+\infty} \frac{\sin \sqrt x}{x-\ln x}\, dx$$ My general ideas are the following:

• for absolute convergence I should use comparison test and therefore find second function but my attempts didn't work.
• for conditional convergence I should use Dirichlet's test where I take $$f(x) = \sin \sqrt x$$ and $$g(x) = \frac{1}{x-\ln x}$$. Are all conditions to use the test met?
• $f(x)$ is not a decreasing sequence, so I don't think Dirichlet's test will work. – D.B. Apr 11 at 21:33
• You have $1/(x - \ln x) \to 0$ monotonically, but $\int_1^x \sin t^{1/3} \, dt$ is not bounded for all $x > 1$ so the Dirichlet test will not work directly. – RRL Apr 12 at 0:19

$$\int_1^c \frac{\sin x^{1/3}}{x- \ln x}dx = \int_1^{c^{1/3}} \frac{\sin u}{u^3- 3\ln u}3u^2 \, du = 3\int_1^{c^{1/3}} \frac{\sin u}{u- 3u^{-2}\ln u} \, du$$
The integral on the RHS is convergent by Dirichlet's test since $$(u- 3u^{-2}\ln u)^{-1} \to 0$$ monotonically as $$u \to \infty$$.
To prove that the integral is not absolutely convergent exploit the periodicity of $$\sin$$ by decomposing into a divergent sum of integrals over intervals of the form $$[k\pi, (k+1)\pi]$$. Proceed as is done for the well-known proof that $$\int_1^\infty x^{-1} \sin x \, dx$$ is not absolutely convergent.