This is a question I met in Fourier analysis chapter in the Mathematic Analysis.
Question. Calculate the limit $$\operatorname*{lim}_{\lambda\to\infty}\int_{0}^{1}\mathrm{ln}x\cos^{2}(\lambda x)\mathrm{d}x$$
I think it must be related to the Riemann lemma which is known as:
Assume $f(x)$ is Riemann integrable or improper intergrable in $[a,b]$, then we have $$\lim_{\lambda\to+\infty}\int_{a}^{b}f\left(x\right)\cos(\lambda x)\mathrm{d}x=0$$
But obviously the question is not satisfied the Riemann lemma's condition, so how to transform the question to satisfy the condition or is there another approach to calculate this limit?
Hint: 1) $\cos^{2}(\lambda x)=\frac {1+\cos (2\lambda x)} 2$. 2) An anti-derivative of $\ln x$ is $x \ln x -x$.