Bourbaki

Exercise 34, § 1

Functions of a Real Variable, III, 124

a) Show that for α>0\alpha > 0 the derivative of

I(α)=∫0+∞e−αxsin⁡xxdx I(\alpha) = \int_0^{+\infty} e^{-\alpha x} \frac{\sin x}{x} dx
is equal to −∫0+∞e−αxsin⁡x dx-\int_0^{+\infty} e^{-\alpha x} \sin x \, dx.

b) Deduce that ∫0+∞sin⁡xxdx=π2\int_0^{+\infty} \frac{\sin x}{x} dx = \frac{\pi}{2}.

Solution

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Transcribed from the printed volume. Not translated.