Do I just straight up integrate (x + cosx)? <--- yes
and to change my interval do I plug Pi/2 and Pi/3 to cos(x)??<--- no, you have to replace x by pi/2 and pi/3 respectively
.....
Spoiler:
You should come out with
$\displaystyle \int_{\frac \pi3}^{\frac \pi2}(x+\cos(x)) dx = \left[\frac12 x^2 + \sin(x) \right]_{\frac \pi3}^{\frac \pi2}= \left(\frac{\pi^2}8+1\right) - \left(\frac{\pi^2}{18}+\frac12 \cdot \sqrt{3} \right)$
awesome... so we didn't have to change the interval of Pi/2 and Pi/3, because usually that's what I'm used to doing..... So since we can't further simplify this equation we just leave it as it is??
Sometimes when I'm doing a substitution on a definite integral, I'll write $\displaystyle \int_{x=a}^{x=b}f(x)\ dx$ so that I remember to change the limits of integration when I do the substitution.