Assume $b>0$. Show that the Lebesgue integral $\int^{\infty}_{1} x^{-b} e^{b\sin(x)} \sin(2x) dx$ exists iff $b>1$.

Question

Assume $b>0$. Show that the Lebesgue integral $\int^{\infty}_{1} x^{-b} e^{b\sin(x)} \sin(2x) dx$ exists iff $b>1$.

Progress: I canton when $b>1$ the integral really is finite but don't know how to show the reverse direction. Any hints would be appreciated.

Answer 1

Hint.

$$\sum_{k=1}^{+\infty} \int_{2k\pi +\frac{\pi}{8}}^{2k\pi +\frac{3\pi}{8}} e^{b \sin x}\frac{\sin 2x}{x^b}dx$$ diverges, by comparison with the harmonic series.

So the function is not Lebesgue integrable.