Definite integration vs differentiation

Question

I have a maybe obvious question is this true?

$$ r = \int_0^{2\pi} \mathrm{d}\theta f(\theta) e^{i\theta} \implies \frac{\mathrm{d}}{\mathrm{d} \theta}(r) = 0 $$

Or should I differentiate first, and then do the definit integral?

Answer 1

The definite integral $r = \int_0^{2\pi}\mathrm{d}\theta f(\theta) e^{i\theta}$ is a number provided that it's defined.

$\dfrac{d}{d\theta} r = 0$ because you are differentiating a number $r$ that doesn't change with $\theta$ with respect to a variable $\theta$.

Answer 2

Since that integral doesn't depend upon $r$, if you differentiate it with respect to $r$, you ger $0$.