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?
Best Answer
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$$\int\sin u\mathrm du=-\cos u+C=-\cos(9+3\sqrt[3]{x})+C.$$