The integral `I=int_(0)^(100pi)[tan^(-1)x]dx` (where, `[.]` represents the greatest integer function) has the vlaue `K(pi)+ tan(p)` then value of `K + p` is equal to

Options

A. 101
B. 99
C. `100pi`
D. `99pi`

Answer

To solve the integral \( I = \int_{0}^{100\pi} [\tan^{-1} x] \, dx \), where \([.]\) represents the greatest integer function, we can break down the problem step by step.

### Step 1: Understanding the Function
The function \(\tan^{-1} x\) (the inverse tangent function) has a range of \((-\frac{\pi}{2}, \frac{\pi}{2})\). As \(x\) approaches \(0\), \(\tan^{-1} x\) approaches \(0\), and as \(x\) approaches \(+\infty\), \(\tan^{-1} x\) approaches \(\frac{\pi}{2}\).

### Step 2: Identifying the Intervals
The greatest integer function \([\tan^{-1} x]\) will take on different integer values depending on the value of \(x\):
- For \(0 \leq x < \tan(1) \approx 1.5574\), \([\tan^{-1} x] = 0\).
- For \(\tan(1) \leq x < \tan(\frac{\pi}{2})\), \([\tan^{-1} x] = 1\).

### Step 3: Breaking the Integral
We can break the integral into two parts:
\[ I = \int_{0}^{\tan(1)} [\tan^{-1} x] \, dx + \int_{\tan(1)}^{100\pi} [\tan^{-1} x] \, dx \]

### Step 4: Evaluating the First Integral
For the first integral, since \([\tan^{-1} x] = 0\) in the interval \(0 \leq x < \tan(1)\):
\[ \int_{0}^{\tan(1)} [\tan^{-1} x] \, dx = \int_{0}^{\tan(1)} 0 \, dx = 0 \]

### Step 5: Evaluating the Second Integral
In the interval \([\tan(1), 100\pi]\), \([\tan^{-1} x] = 1\):
\[ \int_{\tan(1)}^{100\pi} [\tan^{-1} x] \, dx = \int_{\tan(1)}^{100\pi} 1 \, dx = (100\pi - \tan(1)) \]

### Step 6: Combining the Results
Thus, the value of the integral \(I\) is:
\[ I = 0 + (100\pi - \tan(1)) = 100\pi - \tan(1) \]

### Step 7: Expressing in the Given Form
We are given that \(I = K\pi + \tan(p)\). From our result:
\[ 100\pi - \tan(1) = K\pi + \tan(p) \]
This implies:
- \(K = 100\)
- \(\tan(p) = -\tan(1)\)

### Step 8: Finding \(p\)
From \(\tan(p) = -\tan(1)\), we can conclude:
\[ p = -1 \]

### Step 9: Calculating \(K + p\)
Now, we need to find \(K + p\):
\[ K + p = 100 + (-1) = 99 \]

### Final Answer
Thus, the value of \(K + p\) is:
\[ \boxed{99} \]