The problem is:
Judge whether or not the $\int_{1}^{\infty}\ln (\cos\frac{1}{x}+\sin\frac{1}{x}) \Bbb{d}x$ convergent.
I substitute the $\frac{1}{x}$ with $t$ and get $$-\int_{0}^{1} \frac{\ln (\cos \frac{1}{t}+\sin \frac{1}{t})}{t^2}\Bbb{d}t,$$ but I don't know what to do next.
The substitution is used to make the integral simpler, but yours make it more difficult. Consider the integral by part, $$ \int_{1}^{\infty}\ln \left(\cos\frac{1}{x}+\sin\frac{1}{x}\right) \ dx = x\ln \left(\cos\frac{1}{x}+\sin\frac{1}{x}\right)\Bigg|_1^\infty-\int_{1}^{\infty} \frac{\sin\frac{1}{x}-\cos\frac{1}{x}}{x^2\left(\sin\frac{1}{x}+\cos\frac{1}{x}\right)} \ dx $$ and see what you can conclude from this.