For what nonnegative integer $n$ and positive real $c$ does the integral $\int_{1}^{\infty} \ln(1+ \frac{(\sin x)^n}{x^c})dx$ exist as a finite Lebesgue integral and when does it converge as an improper Riemann integral.
Comment: Since there are two parameters, should I fix $n$ or $c$ first?
Monster edit: Sorry, I just realized I found a better starting point.
So, first let's note that for two measurable functions $f_k,k\geq 1$ and $g$ with $f_k \rightarrow g$ it holds
$$\lim_{k\rightarrow \infty}\int f_k(x)dx = \int g(x)dx$$
We are interested in applying this statement to the logarithm function. What can we now take as $f_k(x)$? Looking into Wikipedia gives you the following identity due to Euler:
$$ ln(x)=\lim_{n\rightarrow \infty}n(x^{1/n}-1).$$
Or in our situation $$\int \ln(1+\frac{\sin(x)^n}{x^c}) dx=\lim_{m\rightarrow\infty}\int m\left(\left(1+\frac{\sin(x)^n}{x^c}\right)^{1/m}-1\right) dx. $$
Now, one should continue by checking whether the integrand of the right hand side converges absolutely but appropriately approximating it.