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A $5$-point sequence $x[n]$ is given as

\[ x[-3]=1, \quad x[-2]=1, \quad x[-1]=0, \quad x[0]=5, \quad x[1]=1 .\]

Let $X\left(e^{j \omega}\right)$ denote the discrete-time Fourier transform of $x[n]$. The value of $\int_{-\pi}^{\pi} X\left(e^{j \omega}\right) d \omega$ is

- $5$
- $10 \pi$
- $16 \pi$
- $5+j 10 \pi$