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Mathematics
Calculus
Is the Dirichlet integral a shortcut for solving this difficult integral?
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[QUOTE="Mr Davis 97, post: 6007310, member: 515461"] I have the integral ##\displaystyle \int_{- \infty}^{\infty} \frac{\cos x}{x^2+1} dx##. We are going to use differentiation under the integral sign, so we let ##\displaystyle I(t) = \int_{- \infty}^{\infty} \frac{\cos tx}{x^2+1} dx##, and then, after manipulation, ##\displaystyle I'(t) = \int_{- \infty}^{\infty} \frac{\sin tx}{x(x^2+1)} dx - \int_{- \infty}^{\infty} \frac{\sin tx}{x} dx##. My question lies in the rightmost integral. In a solution I've seen, the rightmost integral is linked to the Dirichlet integral: [URL]https://en.wikipedia.org/wiki/Dirichlet_integral[/URL]. And so ##\pi## is simply substituted for this expression. What I don't understand is how can it be linked to this known integral when there is that ##t## in the argument of ##\sin##? [/QUOTE]
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Forums
Mathematics
Calculus
Is the Dirichlet integral a shortcut for solving this difficult integral?
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