Expert answers to "Integral ∫0ydxsin⁡(ax+bx)x"

siutulysr5

Answered question

2022-01-24

Integral

\int_{0}^{y} d x \frac{\sin (a x + \frac{b}{x})}{x}

Allison Compton

Beginner2022-01-25Added 16 answers

I would enjoy to have a special function for that.
The only thing I have in mind is to simplify the problem using

x = \frac{y}{a}

and

c = a b

to make

\int \frac{\sin (a x + \frac{b}{x})}{x} d x = \int \frac{\sin (y + \frac{c}{y})}{y} d y

and then expand the integrand as a series built at c=0 to get

\frac{\sin (y + \frac{c}{y})}{y} = \sum_{n = 0}^{\infty} \frac{\sin (y + n \frac{π}{2})}{y^{n + 1} n!} c^{n}

and then use

\int \frac{\sin y}{y^{k}} d y = - {\frac{12}{y}}^{- k} ({(- i y)}^{k} Γ (1 - k, - i y) + {(i y)}^{k} Γ (1 - k, i y))

\int \frac{\cos y}{y^{k}} d y = - \frac{12}{i} y^{- k} ({(- i y)}^{k} Γ (1 - k, - i y) - {(i y)}^{k} Γ (1 - k, i y))

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