How to calculate $\int_{e^{-2n\pi }}^{1}\left | \frac{d }{dx}\cos\left ( \ln\frac{1}{x} \right ) \right |dx$

Question

I have no idea on how to calculate $$\int_{e^{-2n\pi }}^{1}\left | \frac{d }{dx}\cos\left ( \ln\frac{1}{x} \right ) \right |dx$$ any hint would be appreciate.

Answer 1

Hint: $$\mathcal{I}=\int_{e^{-2n\pi }}^{1}\left | \frac{\mathrm{d} }{\mathrm{d}x}\cos\left ( \ln\frac{1}{x} \right ) \right |\mathrm{d}x=\int_{e^{-2n\pi }}^{1}\left | \frac{\mathrm{d} }{\mathrm{d} x}\cos\left ( \ln x \right ) \right |\, \mathrm{d}x=\int_{e^{-2n\pi }}^{1}\Big | \sin\left ( \ln x \right ) \Big |\frac{1}{x}\, \mathrm{d}x$$ then let $\ln x=t$ we get $$\mathcal{I}=\int_{-2n\pi }^{0}\Big | \sin t \Big |\, \mathrm{d}t=\int_{0}^{2n\pi }\Big | \sin t \Big |\, \mathrm{d}t$$ hope you can take it from here.