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A weighted sinc function integral.
[Click here for a PDF version of this post] Here’s another real integral problem from [1] (31(f)). Find \begin{equation}\label{eqn:weightedSinc:20} I = \int_0^\infty \frac{\sin x dx}{x\lr{ x^2 + a^2 }}. \end{equation} Both Mathematica and the text state that the answer is \begin{equation}\label{eqn:weightedSinc:40} I = \frac{\pi}{2 a^2} \lr{ 1 – e^{-a}}. \end{equation} My initial attempt to…
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Evaluating some real trig integrals with unit circle contours.
[Click here for a PDF version of this post] Here are a couple of problems from [1]. A sine integral. This is problem (31(a)). For \( a > b > 0 \), find \begin{equation}\label{eqn:unitCircleContourIntegrals:20} I = \int_0^{2 \pi} \frac{d\theta}{a + b \sin\theta}. \end{equation} We can proceed by making a change of variables \( z = e^{i\theta} \), for which \( dz = i z d\theta \). Also let \(…
