Does the integral $ \int \frac{\sin (x)}{x} \ dx $ exist?

Question

I am trying to find the integral,

$$ \int \frac{\sin (9 + 3 {\sqrt[3] {\ x} )}} {\sqrt[3]{\ x^2}} \ dx.$$

I have used a substitution with $u = {(9 + 3 {\sqrt[3] {\ x} )}}$ and $du = dx/\sqrt[3]{\ x^2}$ and that lead me to:

$$ \int \frac{\sin x}{x}\ dx.$$

I want to simplify this further, but am unsure how to do proceed. Does this integral exist?

Answer 1

At @A-LevelStudent's suggestion, I'm noting in this answer that, quite separate from the discussion the OP encouraged (and which other answers provided) of $\int\frac{\sin u}{u}\mathrm du$, the given substitution instead writes the original integral as$$\int\sin u\mathrm du=-\cos u+C=-\cos(9+3\sqrt[3]{x})+C.$$

Answer 2

I'm going to assume you are dealing with real functions here, though it is possible to extend this treatment to the complex domain. The definite integral of the sinc function is the sine integral:

$$\text{Si}(x) = \int \limits_0^x \frac{\sin t}{t} \ dt \quad \quad \quad \text{for } x \in \mathbb{R}.$$

This cannot be written as a finite set of operations on elementary functions, so you are probably stuck with the integral representation, or other representations that are equally complicated. You can find expansions for the sine integral in Harris (2000), but this is fairly complicated. You can also find some results showing uniform convergence of generalised sine integrals in Móricz (2009).

Answer 3

$$\int \frac{\sin(x)}{x}dx=\text{Si}(x)+\text{constant}$$ Si$(x)$ is a known function. One can find its properties in the handbooks of standard functions. For example, see :

https://mathworld.wolfram.com/SineIntegral.html

https://en.wikipedia.org/wiki/Trigonometric_integral#Sine_integral

This is just like $\int \frac{1}{x}dx$ :

As you are familiar with the function $\ln(x)$ you know that $\int \frac{1}{x}dx=\ln(x)+\text{constant}$.

When you will be familiar with the function Si$(x)$ you will know that $\int \frac{\sin(x)}{x}dx=\text{Si}(x)+\text{constant}$.

That may come as a surprise to people who are not familiar with special functions. A discussion about the use of special functions (pp.18-36) : https://fr.scribd.com/doc/14623310/Safari-on-the-country-of-the-Special-Functions-Safari-au-pays-des-fonctions-speciales . The function Si$(x)$ is mentioned in section 4 page 24 and in table 4 page 32 among some special functions on similar kind.