Primitive of the function $(\sin x)/x$

Question

I know that for some functions, for instance $f(x) = e^{-x^2}$, there does not exist a primitive.

Does there is a primitive for the function $f(x) = \frac{\operatorname{sin}(x)}{x}$?

Answer 1

No It is not In terms of elementary functions, But we can write in Infinite Series form.

Using $$\displaystyle \sin x= x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+..........$$

So $$\displaystyle \frac{\sin x}{x} = 1-\frac{x^2}{3!}+\frac{x^4}{4!}-\frac{x^6}{6!}.........$$

So $$\displaystyle \int \frac{\sin x}{x}dx = \int \left\{1-\frac{x^2}{3!}+\frac{x^4}{4!}-\frac{x^6}{6!}.........\right\}dx$$