How could the following integral be solved in a good manner?
$$\int \frac{\sin(x)}{x}\;\mathrm{d}x$$
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This is three elementary integrals in an hour. Is there some reason you are unable to do your own work? – Eric Towers Jul 12 '14 at 03:07
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This doesn't have an elementary anti-derivative. – Jul 12 '14 at 03:10
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No just for learning i have three problem that mentioned all – Nicklaus Josef Jul 12 '14 at 03:10
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@T.Bongers: True. Perhaps "three integrals with elementary integrands." But that's a mouthful that doesn't justify the circumlocution. – Eric Towers Jul 12 '14 at 03:11
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1See http://math.stackexchange.com/questions/163305/what-is-the-integral-of-function-fx-sin-x-x – Eric Towers Jul 12 '14 at 03:14
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This is the definition of function $Si(x)$ (the sine integral) – Claude Leibovici Jul 12 '14 at 03:17
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@ClaudeLeibovici: Almost. Up to a "$+C$" since $\mathrm{Si}(x)$ is the incomplete integral starting at zero. But this is a triviality... – Eric Towers Jul 12 '14 at 03:19
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@EricTowers. Perfectly correct ! Thanks for precising. Cheers :) – Claude Leibovici Jul 12 '14 at 03:20
1 Answers
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In this case, $sin(x)/x$ is harder to tackle, thus, we can only approximate. We have to rely on sin(x)'s power series,
$$\frac {\sin(x)}{x} \approx 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!}$$
$$\int \frac{\sin(x)}{x} dx \approx \int1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} dx$$
$$\int \frac{\sin(x)}{x} dx \approx x - \frac{x^3}{3*3!} + \frac{x^5}{5*5!} - \frac{x^7}{7*7!} + C$$
JoeyAndres
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I thought the power series of $\frac {\sin x}{x} = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} ..$?? – Ishfaaq Jul 12 '14 at 03:22
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