Why some say that $\displaystyle\int\dfrac{\sin(x)}{x}dx$ has no elementary solution? Because I can solve it using only elementary ways:
$$\int\dfrac{\sin(x)}{x}dx=\int\sin(x)x^{-1}~dx=\sin(x)\ln(x)-\int\cos(x)\ln(x)~dx$$
and then I try to find somewhy but it is elementary.
I think people misinterpreted the OP's question. I won't explain what I think the OP means, my answer should make that clear.
You're using the term 'elementary' in a different way than what it is meant.
When people say that '$\displaystyle \int\dfrac{{\sin{(x)}}}{{x}}\mathrm dx$ has no elementary solution', what is meant isn't that an antiderivative of $x\mapsto \dfrac{{\sin{(x)}}}{{x}}$ can't be found using only elementary techniques, but rather that an antiderivative of $x\mapsto \dfrac{{\sin{(x)}}}{{x}}$ can't be expressed in terms of elementary functions.
What you suggest may, a priori, have a chance of being an elementary way of finding an antiderivative (I doubt it), but what you'll find in the end won't be an elementary function.