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Is it true that $$\int_0^\infty\frac{|\cos(x)|}{x+1} dx$$ diverges?

Any proofs?


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1 Answer

You can use the fact that $\int_{n\pi}^{(n+1)\pi} |\cos x| \,\mathrm{d}x = 2$.

This implies that $\int_{n\pi}^{(n+1)\pi} \frac{|\cos x| \,\mathrm{d}x}{x+1} \ge \int_{n\pi}^{(n+1)\pi} \frac{|\cos x| \,\mathrm{d}x}{(n+2)\pi} = \frac{2}{(n+2)\pi}$.

Hence $$\int_{0}^{\infty} \frac{|\cos x| \,\mathrm{d}x}{x+1} \ge \frac1\pi \sum_{k=0}^\infty \frac1{k+2}=\infty.$$

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