For which values $p \in [1, \infty)$ does the function $f(x)=\frac{\ln(1+3x)}{\sqrt{x} \sin ^2 x}$ belong to space $L^p ((0,1), \tan x\, dx)$ ?
So, equivalently, we have to find all the $p$ values so the integral $\int_{0}^{1} \frac{\ln^p(1+3x)}{{x}^{\frac{p}{2}} \sin ^{2p} x} \tan x $ can converge. Can anyone help me prove this? I guess the easiest way would be using the comparison test, but I couldn't find any appropriate function. The proof for $p=1$ was easy and I did it. Thanks
If $x$ is close to zero then $\sin x \sim x$, $\tan x \sim x$, and $\ln (1+3x) \sim 3x$.
You can write $$ \frac{\ln^p(1+3x) \tan x}{x^{p/2} \sin^{2p}x} =3^p\frac{\ln^p(1+3x)}{(3x)^p} \frac{\tan x}{x} \cdot\frac{x^{2p}} {\sin^{2p}x}\cdot x^{1 - p - p/2} \sim 3^px^{1-p-p/2}. $$
The factor of $3^p$ is immaterial. The function $x^{1-p-p/2}$ is integrable near zero if and only if $1 - p - p/2 > -1$.