How would you integrate the trignometric integral function Si(x)?

Question

The function Si(x) can be obtained when we integrate $\frac {sin(x)}x$. But how would we go about integrating Si(x)?

More information about the function Si(x) can be found here https://en.wikipedia.org/wiki/Trigonometric_integral

Answer 1

This can be done analytically. You have $$\DeclareMathOperator{\Si}{Si} \int \Si(x)\,dx = x \Si(x) + \cos(x) + \text{constant} $$ Test by differentiating the RHS: $$ \frac{d}{dx} (x \Si(x) + \cos(x) + \text{constant}) = \Si(x) + x \frac{\sin x}{x} - \sin(x) = \Si(x) $$ If you couldn't guess the result, you can get it by integrating by parts: $$ \int \Si(x)\,dx =\int 1 \cdot \Si(x)\,dx = x \Si(x) - \int x \frac{\sin x}{x} = x \Si(x) + \cos x + \text{constant} $$