How Is the Integral of $\frac{\sin x}{x}$ Related to the Sine Integral Function?

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SUMMARY

The integral of $\frac{\sin x}{x}$ cannot be expressed using elementary functions. Instead, it is defined in terms of the sine integral function, denoted as Si(x). The relationship is established as follows: $\int \frac{\sin x}{x} dx = Si(x)$, which is the precise definition of the sine integral function.

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[tex]\int \frac{\sin x}{x }dx[/tex]
 
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That anti-derivative cannot be written in terms of elementary functions. It can be written easily in terms of the "sine integral" function, Si(x):
[tex]\int \frac {sin(x)}{x} dx= Si(x)[/tex]
because that is exactly how Si(x) is defined!
 

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