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I'm trying to figure out how to express the integral $$\int\limits_{0}^{\infty} \cos(x) \times x^{-a} \rm{dx}$$ as

$$\cos\frac{\pi-a\pi}{2}\times \int\limits_{0}^{\infty} e^{-x} x^{-a} \ \rm{dx} \qquad \text{where} \ 0 < a <1$$

I'm fairly sure it requires the calculation of a residue, and I've tried using a Fourier transform but it doesn't seem to get me anywhere. Sorry about the poor formatting, it's my first time posting and I can't figure out how to express the integrals nicely.

Any hints on how to proceed?

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