Evaluate expressions without computing DTFT Let $X(\omega)$ be the DTFT of the sequence $x[n]$ given by:
$$
x[n] = \{4, 2, -1, 5, -3, 1, -2, 4, 2\},\quad\text{for}\quad n \in [-6, 2]
$$
I do want to compute

$X(0)$
$X(\pi)$
$\displaystyle\int_{-\pi}^{\pi} X(\omega) d{\omega}$
$\displaystyle\int_{-\pi}^{\pi}|X(\omega)|^2 d{\omega}$
$\displaystyle\int_{-\pi}^{\pi} \bigg\lvert\frac{dX(\omega)}{d{\omega}}\bigg\rvert^2 d{\omega}$

What I have tried

For $X(0)$, I computed the sum and so it was 12.
For $X(\pi)$ I did $X(\pi) = x[9] = \sum_{n = 0}^{8} x[n]e^{-i\omega n} = (-1)^9 \times 12 = -12$, because 0 corresponds to 0, 2 to $\pi/8$, -1 to $\pi/4$, ..., 2 to $\pi$.
For $\int_{-\pi}^{\pi} X(\omega) d{\omega}$, I tried by Parseval Theorem, so it was $2\pi x[n] x[n]*$, but I don't know if it's correct. Last two I don't have any idea of how to do it.

I also have a similar exercise, but it is with DFT instead of DTFT, but it is pretty the same, isn't it?
Thanks in advance.
For the first two questions you just need to use the definition of the DTFT
$$X(\omega)=\sum_{n=-\infty}^{\infty}x[n]e^{-jn\omega}\tag{1}$$
and use $\omega=0$ and $\omega=\pi$, respectively.
For 3. you just need to use the definition of the inverse DTFT:
$$x[n]=\frac{1}{2\pi}\int_{-\pi}^{\pi}X(\omega)e^{jn\omega}d\omega\tag{2}$$
Finally, for 4. and 5. you need Parseval's theorem. In addition, for 5. you need to express the sequence corresponding to the derivative of $X(\omega)$ in terms of $x[n]$. You should be able to either derive this yourself or look it up in a table.