# Integrate sin(u)/u with respect to u (related to the Sine integral Si(x)?

using integration by parts,
Int [(sin u)/u] du = sin u * ln u - Int cos u * ln u du
Int cos u ln u du = sin u ln u - Int (sin u)/u du
So
Int [(sin u)/u] du = sin u * ln u - sin u ln u + Int (sin u)/u du
or 0 = 0.
So this method does not work...
By the way, the Sine integral...
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using integration by parts,

Int [(sin u)/u] du = sin u * ln u - Int cos u * ln u du

Int cos u ln u du = sin u ln u - Int (sin u)/u du

So

Int [(sin u)/u] du = sin u * ln u - sin u ln u + Int (sin u)/u du

or 0 = 0.

So this method does not work...

By the way, the Sine integral Si(x) is defined to be

Si(x) = Int[0 to x] [(sin u)/u] du

Int [(sin u)/u] du = sin u * ln u - Int cos u * ln u du

Int cos u ln u du = sin u ln u - Int (sin u)/u du

So

Int [(sin u)/u] du = sin u * ln u - sin u ln u + Int (sin u)/u du

or 0 = 0.

So this method does not work...

By the way, the Sine integral Si(x) is defined to be

Si(x) = Int[0 to x] [(sin u)/u] du

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