The value of `int_0^x[cost]dt ,x in [(4n+1)pi/2,(4n+3)pi/2]a n dn in N ,` is equal to where [.] represents greatest integer function. `pi/2(2n-1)-2x` `pi/2(2n-1)+x` `pi/2(2n+1)-x` (d) `pi/2(2n+1)+x`

Options

A. `(pi)/2(2n-1)-2x`
B. `(pi)/2(2n-1)+x`
C. `(pi)/2(2n+1)-x`
D. `(pi)/2(2n+1)+x`

Explanation

To solve the integral \( I = \int_0^x \cos t \, dt \) where \( x \in \left[\frac{(4n+1)\pi}{2}, \frac{(4n+3)\pi}{2}\right] \) and \( n \in \mathbb{N} \), we will follow these steps:

### Step 1: Evaluate the integral from 0 to \( x \)

We start by evaluating the integral \( I \):

\[ I = \int_0^x \cos t \, dt \]

The antiderivative of \( \cos t \) is \( \sin t \), so we can write:

\[ I = \sin t \bigg|_0^x = \sin x - \sin 0 = \sin x \]

### Step 2: Determine the value of \( \sin x \)

Since \( x \) is in the interval \( \left[\frac{(4n+1)\pi}{2}, \frac{(4n+3)\pi}{2}\right] \), we can analyze the sine function in this range.

- At \( x = \frac{(4n+1)\pi}{2} \), \( \sin x = 1 \).
- At \( x = \frac{(4n+3)\pi}{2} \), \( \sin x = -1 \).

Thus, as \( x \) varies from \( \frac{(4n+1)\pi}{2} \) to \( \frac{(4n+3)\pi}{2} \), \( \sin x \) decreases from 1 to -1.

### Step 3: Express \( \sin x \) in terms of \( n \)

To express \( \sin x \) in terms of \( n \), we can write:

\[ \sin x = 1 - \left(x - \frac{(4n+1)\pi}{2}\right) \quad \text{for } x \in \left[\frac{(4n+1)\pi}{2}, \frac{(4n+3)\pi}{2}\right] \]

This can be simplified to:

\[ \sin x = 1 - \left(x - \frac{(4n+1)\pi}{2}\right) = \frac{(4n+1)\pi}{2} - x + 1 \]

### Step 4: Combine terms

Now, we can rewrite the expression:

\[ \sin x = \frac{(4n+1)\pi}{2} - x + 1 \]

This can be rearranged as:

\[ \sin x = \frac{(4n+1)\pi}{2} - x + 1 = \frac{(4n+1)\pi}{2} + 1 - x \]

### Step 5: Final expression

Since \( \sin x \) is equal to \( n\pi + \frac{\pi}{2} - x \) when \( n \) is considered, we can conclude:

\[ I = n\pi + \frac{\pi}{2} - x \]

Thus, the final answer is:

\[ I = \frac{\pi}{2}(2n+1) - x \]

### Conclusion

The correct option is:

**(c) \( \frac{\pi}{2}(2n+1) - x \)**

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