To solve the integral \(\int_{-\pi}^{5\pi} \cot^{-1}(\cot x) \, dx\), we can follow these steps:
### Step 1: Understand the function \(\cot^{-1}(\cot x)\)
The function \(\cot^{-1}(\cot x)\) simplifies based on the properties of the cotangent and the inverse cotangent functions. The value of \(\cot^{-1}(\cot x)\) is defined as:
- \(x\) if \(x \in (0, \pi)\)
- \(x - \pi\) if \(x \in (\pi, 2\pi)\)
- \(x - 2\pi\) if \(x \in (2\pi, 3\pi)\)
- and so on...
This means that \(\cot^{-1}(\cot x)\) is periodic with a period of \(\pi\) and takes values in the interval \((0, \pi)\).
### Step 2: Break the integral into intervals
Given the periodicity of \(\cot^{-1}(\cot x)\), we can break the integral from \(-\pi\) to \(5\pi\) into smaller intervals of length \(\pi\):
\[
\int_{-\pi}^{5\pi} \cot^{-1}(\cot x) \, dx = \int_{-\pi}^{0} \cot^{-1}(\cot x) \, dx + \int_{0}^{\pi} \cot^{-1}(\cot x) \, dx + \int_{\pi}^{2\pi} \cot^{-1}(\cot x) \, dx + \int_{2\pi}^{3\pi} \cot^{-1}(\cot x) \, dx + \int_{3\pi}^{4\pi} \cot^{-1}(\cot x) \, dx + \int_{4\pi}^{5\pi} \cot^{-1}(\cot x) \, dx
\]
### Step 3: Evaluate each integral
1. **For \([- \pi, 0]\)**:
- Here, \(\cot^{-1}(\cot x) = \pi + x\) since \(x\) is in the range \((- \pi, 0)\).
- Thus, \(\int_{-\pi}^{0} \cot^{-1}(\cot x) \, dx = \int_{-\pi}^{0} (\pi + x) \, dx\).
2. **For \([0, \pi]\)**:
- Here, \(\cot^{-1}(\cot x) = x\).
- Thus, \(\int_{0}^{\pi} \cot^{-1}(\cot x) \, dx = \int_{0}^{\pi} x \, dx\).
3. **For \([\pi, 2\pi]\)**:
- Here, \(\cot^{-1}(\cot x) = x - \pi\).
- Thus, \(\int_{\pi}^{2\pi} \cot^{-1}(\cot x) \, dx = \int_{\pi}^{2\pi} (x - \pi) \, dx\).
4. **For \([2\pi, 3\pi]\)**:
- Here, \(\cot^{-1}(\cot x) = x - 2\pi\).
- Thus, \(\int_{2\pi}^{3\pi} \cot^{-1}(\cot x) \, dx = \int_{2\pi}^{3\pi} (x - 2\pi) \, dx\).
5. **For \([3\pi, 4\pi]\)**:
- Here, \(\cot^{-1}(\cot x) = x - 3\pi\).
- Thus, \(\int_{3\pi}^{4\pi} \cot^{-1}(\cot x) \, dx = \int_{3\pi}^{4\pi} (x - 3\pi) \, dx\).
6. **For \([4\pi, 5\pi]\)**:
- Here, \(\cot^{-1}(\cot x) = x - 4\pi\).
- Thus, \(\int_{4\pi}^{5\pi} \cot^{-1}(\cot x) \, dx = \int_{4\pi}^{5\pi} (x - 4\pi) \, dx\).
### Step 4: Calculate each integral
1. \(\int_{-\pi}^{0} (\pi + x) \, dx = \left[ \pi x + \frac{x^2}{2} \right]_{-\pi}^{0} = \left(0 - \left(-\pi^2 + \frac{\pi^2}{2}\right)\right) = \frac{\pi^2}{2}\)
2. \(\int_{0}^{\pi} x \, dx = \left[ \frac{x^2}{2} \right]_{0}^{\pi} = \frac{\pi^2}{2}\)
3. \(\int_{\pi}^{2\pi} (x - \pi) \, dx = \left[ \frac{x^2}{2} - \pi x \right]_{\pi}^{2\pi} = \left(\frac{(2\pi)^2}{2} - 2\pi^2\right) - \left(\frac{\pi^2}{2} - \pi^2\right) = \frac{\pi^2}{2}\)
4. \(\int_{2\pi}^{3\pi} (x - 2\pi) \, dx = \frac{\pi^2}{2}\)
5. \(\int_{3\pi}^{4\pi} (x - 3\pi) \, dx = \frac{\pi^2}{2}\)
6. \(\int_{4\pi}^{5\pi} (x - 4\pi) \, dx = \frac{\pi^2}{2}\)
### Step 5: Sum all the integrals
Now, we can sum all the results:
\[
\int_{-\pi}^{5\pi} \cot^{-1}(\cot x) \, dx = \frac{\pi^2}{2} + \frac{\pi^2}{2} + \frac{\pi^2}{2} + \frac{\pi^2}{2} + \frac{\pi^2}{2} + \frac{\pi^2}{2} = 6 \cdot \frac{\pi^2}{2} = 3\pi^2
\]
### Final Result
Thus, the value of the integral is:
\[
\int_{-\pi}^{5\pi} \cot^{-1}(\cot x) \, dx = 3\pi^2
\]
