What's the way to calculate the following integral $$\int_0^{\infty}\frac{\cos x-e^{-x}}{x}\,dx?$$ The answer is zero. I tried to split into two integrals $\int_0^{\infty}\frac{1-\cos x}{x}\,dx$ and $\int_0^{\infty}\frac{1-e^{-x}}{x}\,dx$ but they both don't exist. Integration by parts failed for the same reason: integral and one of the limits don't exist.
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