How to integrate $\frac{\cos(x)}{x}$ using substitution

Question

Trying to integrate $$\int \frac{\cos(x)}{x} dx = \int \frac{1}{x}\sin'(x) dx$$ by substituting $\sin(x)$, but it either becomes more complicated or I end up with a $\frac{1}{x}$ still in the integral.

Answer 1

It is not possible to find an antiderivative of $\frac{\cos x}{x}$ in term of "elementary functions".

This is a consequence of Liouville's theorem. See link to article for details.

Answer 2

As noted the indefinite integral $$ \int \frac{\cos x}{x}\;dx $$ is not an elementary function. But it is useful enough that it has been given a name, the "cosine integral" function, $\mathrm{Ci}(x)$. It is conventional to fix the constant of integration so that $\lim_{x \to +\infty} \mathrm{Ci}(x) = 0$. So we may define $$ \mathrm{Ci}(x) = -\int_x^\infty\frac{\cos t}{t}\;dt $$ In fact, this definition makes sense for $x$ in the complex plane, with a cut along the negative real axis.