The value of `int_(1)^(4){x}^([x]) dx` (where , [.] and {.} denotes the greatest integer and fractional part of x) is equal to

Options

A. (a)`(11)/(12)`
B. (b)`(13)/(12)`
C. (c)`(70)/(12)`
D. (d)`(19)/(12)`

Answer

To solve the integral \( \int_{1}^{4} x^{[x]} \, dx \), where \([x]\) is the greatest integer function and \(\{x\}\) is the fractional part of \(x\), we will break the integral into segments based on the behavior of the greatest integer function.

### Step 1: Identify the intervals
The greatest integer function \([x]\) changes at integer values. Therefore, we can break the integral from 1 to 4 into three segments:
- From 1 to 2: Here, \([x] = 1\)
- From 2 to 3: Here, \([x] = 2\)
- From 3 to 4: Here, \([x] = 3\)

Thus, we can write:
\[ \int_{1}^{4} x^{[x]} \, dx = \int_{1}^{2} x^{1} \, dx + \int_{2}^{3} x^{2} \, dx + \int_{3}^{4} x^{3} \, dx \]

### Step 2: Solve each integral
1. **Integral from 1 to 2:**
\[ \int_{1}^{2} x^{1} \, dx = \left[ \frac{x^2}{2} \right]_{1}^{2} = \frac{2^2}{2} - \frac{1^2}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2} \]

2. **Integral from 2 to 3:**
\[ \int_{2}^{3} x^{2} \, dx = \left[ \frac{x^3}{3} \right]_{2}^{3} = \frac{3^3}{3} - \frac{2^3}{3} = \frac{27}{3} - \frac{8}{3} = \frac{19}{3} \]

3. **Integral from 3 to 4:**
\[ \int_{3}^{4} x^{3} \, dx = \left[ \frac{x^4}{4} \right]_{3}^{4} = \frac{4^4}{4} - \frac{3^4}{4} = \frac{256}{4} - \frac{81}{4} = \frac{175}{4} \]

### Step 3: Combine the results
Now, we combine the results of the three integrals:
\[ \int_{1}^{4} x^{[x]} \, dx = \frac{3}{2} + \frac{19}{3} + \frac{175}{4} \]

### Step 4: Find a common denominator
The least common multiple of the denominators \(2, 3, 4\) is \(12\). We convert each fraction:
- \(\frac{3}{2} = \frac{18}{12}\)
- \(\frac{19}{3} = \frac{76}{12}\)
- \(\frac{175}{4} = \frac{525}{12}\)

### Step 5: Add the fractions
Now we can add:
\[ \int_{1}^{4} x^{[x]} \, dx = \frac{18}{12} + \frac{76}{12} + \frac{525}{12} = \frac{619}{12} \]

### Final Answer
Thus, the value of the integral is:
\[ \int_{1}^{4} x^{[x]} \, dx = \frac{619}{12} \]

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