`int_- 10^20[cot^(- 1)x]dx`

Answer

To solve the integral \(\int_{-10}^{20} \cot^{-1}(x) \, dx\), we will break down the problem step by step.

### Step 1: Understanding the Function
The function we are integrating is \(\cot^{-1}(x)\), which is the inverse cotangent function. We need to evaluate the definite integral from \(-10\) to \(20\).

### Step 2: Finding the Limits
The limits of integration are from \(-10\) to \(20\). We will first find the values of \(\cot^{-1}(x)\) at the endpoints and any critical points in between.

### Step 3: Identify Critical Points
The function \(\cot^{-1}(x)\) has specific values at certain points:
- \(\cot^{-1}(0) = \frac{\pi}{2}\)
- \(\cot^{-1}(1) = \frac{\pi}{4}\)
- \(\cot^{-1}(2)\)
- \(\cot^{-1}(3)\)

We will evaluate the function at these points to determine the behavior of the function over the interval.

### Step 4: Break the Integral into Parts
To evaluate the integral, we can break it into segments based on the values of \(\cot^{-1}(x)\):
1. From \(-10\) to \(\cot^{-1}(3)\)
2. From \(\cot^{-1}(3)\) to \(\cot^{-1}(2)\)
3. From \(\cot^{-1}(2)\) to \(\frac{\pi}{4}\)
4. From \(\frac{\pi}{4}\) to \(20\)

### Step 5: Evaluate Each Segment
1. **From \(-10\) to \(\cot^{-1}(3)\)**:
- Here, \(\cot^{-1}(x)\) is approximately \(3\), so the integral becomes:
\[ \int_{-10}^{\cot^{-1}(3)} 3 \, dx = 3 \left[ x \right]_{-10}^{\cot^{-1}(3)} = 3 \left( \cot^{-1}(3) + 10 \right) \]

2. **From \(\cot^{-1}(3)\) to \(\cot^{-1}(2)\)**:
- The value of \(\cot^{-1}(x)\) is approximately \(2\):
\[ \int_{\cot^{-1}(3)}^{\cot^{-1}(2)} 2 \, dx = 2 \left[ x \right]_{\cot^{-1}(3)}^{\cot^{-1}(2)} = 2 \left( \cot^{-1}(2) - \cot^{-1}(3) \right) \]

3. **From \(\cot^{-1}(2)\) to \(\frac{\pi}{4}\)**:
- The value of \(\cot^{-1}(x)\) is approximately \(1\):
\[ \int_{\cot^{-1}(2)}^{\frac{\pi}{4}} 1 \, dx = \left[ x \right]_{\cot^{-1}(2)}^{\frac{\pi}{4}} = \frac{\pi}{4} - \cot^{-1}(2) \]

4. **From \(\frac{\pi}{4}\) to \(20\)**:
- The value of \(\cot^{-1}(x)\) is \(0\) for \(x > 0\):
\[ \int_{\frac{\pi}{4}}^{20} 0 \, dx = 0 \]

### Step 6: Combine the Results
Now, we combine all the evaluated segments:
\[ \text{Total} = 3 \left( \cot^{-1}(3) + 10 \right) + 2 \left( \cot^{-1}(2) - \cot^{-1}(3) \right) + \left( \frac{\pi}{4} - \cot^{-1}(2) \right) + 0 \]

### Step 7: Simplify the Expression
Combine like terms:
\[ = 3 \cot^{-1}(3) + 30 + 2 \cot^{-1}(2) - 2 \cot^{-1}(3) + \frac{\pi}{4} - \cot^{-1}(2) \]
\[ = (3 - 2) \cot^{-1}(3) + (2 - 1) \cot^{-1}(2) + 30 + \frac{\pi}{4} \]
\[ = \cot^{-1}(3) + \cot^{-1}(2) + 30 + \frac{\pi}{4} \]

### Final Result
Thus, the final result of the integral is:
\[ \int_{-10}^{20} \cot^{-1}(x) \, dx = \cot^{-1}(3) + \cot^{-1}(2) + 30 + \frac{\pi}{4} \]

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