Is function in $L^p$?

Question

When $\frac{\ln{(1+3x)}}{\sqrt{x}\sin^2{x}}$ is in $L^p((0,1), \tan{x}\,dx)$?

I used that $\ln{(1+3x)}\sim x$, $\sin{x}\sim x$ when $x\to 0$ and integral converges for $3p/2-1<1$ or $p<\frac{4}{3}$. Am I correct?

Answer 1

Yes, you are correct. If $x\to 0^+$ then $$\frac{\ln^p{(1+3x)}}{x^{p/2}\sin^{2p}{x}}\cdot \tan(x)\sim \frac{3^px^p}{x^{p/2+2p}}\cdot x=\frac{3^p}{x^{3p/2-1}}$$ and the integral over $(0,1)$ converges when $3p/2-1<1$, that is $p<4/3$.