If `f(n)=(int_0^n[x]dx)/(int_0^n{x}dx)`(where,[*] and {*} denotes greatest integer and fractional part of x and `n in N).` Then, the value of `f(4)` is...

Answer

To solve the problem, we need to evaluate the function \( f(n) \) defined as:

\[ f(n) = \frac{\int_0^n [x] \, dx}{\int_0^n \{x\} \, dx} \]

where \([x]\) is the greatest integer function and \(\{x\}\) is the fractional part of \(x\).

### Step 1: Evaluate the numerator \(\int_0^n [x] \, dx\)

The greatest integer function \([x]\) takes the following values in the intervals:
- For \(0 \leq x < 1\), \([x] = 0\)
- For \(1 \leq x < 2\), \([x] = 1\)
- For \(2 \leq x < 3\), \([x] = 2\)
- For \(3 \leq x < 4\), \([x] = 3\)
- For \(n\) (where \(n\) is a natural number), \([x] = n-1\) for \(n-1 \leq x < n\)

Thus, we can break the integral into segments:

\[ \int_0^n [x] \, dx = \int_0^1 0 \, dx + \int_1^2 1 \, dx + \int_2^3 2 \, dx + \int_3^4 3 \, dx + \int_4^n (n-1) \, dx \]

Calculating each part:
- \(\int_0^1 0 \, dx = 0\)
- \(\int_1^2 1 \, dx = 1\)
- \(\int_2^3 2 \, dx = 2\)
- \(\int_3^4 3 \, dx = 3\)
- \(\int_4^n (n-1) \, dx = (n-1)(n-4)\)

Adding these together gives:

\[ \int_0^n [x] \, dx = 0 + 1 + 2 + 3 + (n-1)(n-4) = 6 + (n-1)(n-4) \]

### Step 2: Evaluate the denominator \(\int_0^n \{x\} \, dx\)

The fractional part \(\{x\}\) can be expressed as \(\{x\} = x - [x]\). Thus:

\[ \int_0^n \{x\} \, dx = \int_0^n (x - [x]) \, dx = \int_0^n x \, dx - \int_0^n [x] \, dx \]

Calculating \(\int_0^n x \, dx\):

\[ \int_0^n x \, dx = \frac{n^2}{2} \]

Now we already have \(\int_0^n [x] \, dx\) from Step 1, so we can substitute:

\[ \int_0^n \{x\} \, dx = \frac{n^2}{2} - (6 + (n-1)(n-4)) \]

Simplifying the denominator:

\[ \int_0^n \{x\} \, dx = \frac{n^2}{2} - 6 - (n^2 - 5n + 4) = \frac{n^2}{2} - 6 - n^2 + 5n - 4 \]

Combining terms gives:

\[ \int_0^n \{x\} \, dx = -\frac{n^2}{2} + 5n - 10 \]

### Step 3: Combine the results to find \( f(n) \)

Now substituting back into the function \( f(n) \):

\[ f(n) = \frac{6 + (n-1)(n-4)}{-\frac{n^2}{2} + 5n - 10} \]

### Step 4: Evaluate \( f(4) \)

Substituting \( n = 4 \):

Numerator:

\[ 6 + (4-1)(4-4) = 6 + 3 \cdot 0 = 6 \]

Denominator:

\[ -\frac{4^2}{2} + 5 \cdot 4 - 10 = -8 + 20 - 10 = 2 \]

Thus:

\[ f(4) = \frac{6}{2} = 3 \]

### Final Answer:

The value of \( f(4) \) is \( \boxed{3} \).