Evaluate: `int_0^x[cost]dt where x in (2npi,(4n+1pi/2),n in N ,a n d[dot]` denotes the greatest integer function.

Explanation

To evaluate the integral \( I = \int_0^x \lfloor \cos t \rfloor dt \) where \( x \) is in the interval \( (2n\pi, (4n + 1)\frac{\pi}{2}) \) for \( n \in \mathbb{N} \), we will follow these steps:

### Step 1: Understand the behavior of \( \cos t \)
The function \( \cos t \) oscillates between -1 and 1. The greatest integer function \( \lfloor \cos t \rfloor \) will take the following values:
- \( \lfloor \cos t \rfloor = 1 \) when \( \cos t \in [1, 0) \) (i.e., \( t \in [0, \frac{\pi}{2}) \))
- \( \lfloor \cos t \rfloor = 0 \) when \( \cos t \in [0, -1) \) (i.e., \( t \in [\frac{\pi}{2}, \frac{3\pi}{2}) \))
- \( \lfloor \cos t \rfloor = -1 \) when \( \cos t \in (-1, -1) \) (i.e., \( t \in [\frac{3\pi}{2}, 2\pi) \))

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