You have a mistake on the next-to-last line. You have evaluated your antiderivative in the wrong order. Inside the parentheses you should have ln(sin(##\pi/2##)) - ln(sin(##\pi/6##)).
Also, the differential -- dx or du -- should appear in every step until you actually get the antiderivative.
When you make your substitution, that's when you should either change your limits of integration or at least make a note that the limits are values of x, not u. Here's what I mean:
$$ \pi \int_{x = \pi/6}^{\pi/2} \frac{du}{u}$$
By including "x = ..." in one of the limits of integration, that reminds you not to use them directly in the antiderivative that you get.