Find the value of `int_(0)^(100pi)sin^(-1)(sin x)dx`.

Explanation

To solve the integral \( I = \int_{0}^{100\pi} \sin^{-1}(\sin x) \, dx \), we will analyze the behavior of the function \( \sin^{-1}(\sin x) \) over the given interval.

### Step 1: Understanding the Function
The function \( \sin^{-1}(\sin x) \) gives the angle whose sine is \( \sin x \). However, \( \sin^{-1}(y) \) is defined only for \( y \) in the range \([-1, 1]\) and returns values in the interval \([- \frac{\pi}{2}, \frac{\pi}{2}]\). Thus, we need to determine how \( \sin^{-1}(\sin x) \) behaves over the intervals of \( x \).

### Step 2: Analyzing the Periodicity
The sine function is periodic with a period of \( 2\pi \). Therefore, \( \sin^{-1}(\sin x) \) will also exhibit periodic behavior. Specifically:
- For \( x \in [0, \pi] \), \( \sin^{-1}(\sin x) = x \).
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