Word Count : 522

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We are asked to evaluate [ I(x) = \int_0^\infty \frac{\cos(\alpha x) + \alpha \sin(\alpha x)}{1 + \alpha^2} , d\alpha ] using Fourier integral representation. Let's proceed step by step. --- Consider the Fourier integral representation for a function (f(t)): [ f(x) = \frac{1}{\pi} \int_0^\infty F_c(\alpha) \cos(\alpha x), d\alpha + \frac{1}{\pi} \int_0^\infty F_s(\alpha) \sin(\alpha x), d\alpha ] where (F_c(\alpha) = \int_{-\infty}^{\infty} f(t) \cos(\alpha t), dt) and (F_s(\alpha) = \int_{-\infty}^{\infty} f(t) \sin(\alpha t), dt). Notice that the given integral has the form [ I(x) = \int_0^\infty \frac{\cos(\alpha x)}{1+\alpha^2} , _______ ______ ___ ________ _______ ________ __________ ___ __________.
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