Let `[x]` denote the greatest integer `lex`.Consider the function `f(x)=max{x^(2),1+[x]}`.Then the value of the integral `int_(0)^(2)f(x)(dx` is

Options

A. `(5+4sqrt(2))/(3)`
B. `(8+4sqrt(2))/(3)`
C. `(1+5sqrt(2))/(3)`
D. `(4+5sqrt(2))/(3)`

Answer

To solve the integral \( \int_{0}^{2} f(x) \, dx \) where \( f(x) = \max\{x^2, 1 + [x]\} \), we first need to analyze the function \( f(x) \) over the interval from \( 0 \) to \( 2 \).

### Step 1: Determine \( f(x) \) in the interval \( [0, 1) \)

In the interval \( [0, 1) \):
- The greatest integer function \( [x] = 0 \).
- Thus, \( 1 + [x] = 1 + 0 = 1 \).
- Therefore, \( f(x) = \max\{x^2, 1\} \).

Since \( x^2 \) ranges from \( 0 \) to \( 1 \) in this interval, we have:
- For \( 0 \leq x < 1 \), \( f(x) = 1 \).

### Step 2: Determine \( f(x) \) in the interval \( [1, 2) \)

In the interval \( [1, 2) \):
- The greatest integer function \( [x] = 1 \).
- Thus, \( 1 + [x] = 1 + 1 = 2 \).
- Therefore, \( f(x) = \max\{x^2, 2\} \).

Now, we need to find where \( x^2 \) intersects \( 2 \):
- The equation \( x^2 = 2 \) gives \( x = \sqrt{2} \).
- For \( 1 \leq x < \sqrt{2} \), \( x^2 < 2 \) so \( f(x) = 2 \).
- For \( \sqrt{2} \leq x < 2 \), \( x^2 \geq 2 \) so \( f(x) = x^2 \).

### Step 3: Set up the integral

Now we can set up the integral as follows:
\[ \int_{0}^{2} f(x) \, dx = \int_{0}^{1} f(x) \, dx + \int_{1}^{\sqrt{2}} f(x) \, dx + \int_{\sqrt{2}}^{2} f(x) \, dx \]

Calculating each part:
1. **From \( 0 \) to \( 1 \)**:
\[ \int_{0}^{1} f(x) \, dx = \int_{0}^{1} 1 \, dx = [x]_{0}^{1} = 1 - 0 = 1 \]

2. **From \( 1 \) to \( \sqrt{2} \)**:
\[ \int_{1}^{\sqrt{2}} f(x) \, dx = \int_{1}^{\sqrt{2}} 2 \, dx = [2x]_{1}^{\sqrt{2}} = 2\sqrt{2} - 2 \]

3. **From \( \sqrt{2} \) to \( 2 \)**:
\[ \int_{\sqrt{2}}^{2} f(x) \, dx = \int_{\sqrt{2}}^{2} x^2 \, dx = \left[\frac{x^3}{3}\right]_{\sqrt{2}}^{2} = \frac{8}{3} - \frac{(\sqrt{2})^3}{3} = \frac{8}{3} - \frac{2\sqrt{2}}{3} \]

### Step 4: Combine the results

Now, we can combine all parts:
\[ \int_{0}^{2} f(x) \, dx = 1 + (2\sqrt{2} - 2) + \left(\frac{8}{3} - \frac{2\sqrt{2}}{3}\right) \]

Combine the constants and the terms involving \( \sqrt{2} \):
\[ = 1 - 2 + 2\sqrt{2} + \frac{8}{3} - \frac{2\sqrt{2}}{3} \]
\[ = -1 + 2\sqrt{2} + \frac{8}{3} - \frac{2\sqrt{2}}{3} \]
\[ = -1 + \left(2 - \frac{2}{3}\right)\sqrt{2} + \frac{8}{3} \]
\[ = -1 + \frac{6}{3}\sqrt{2} - \frac{2}{3}\sqrt{2} + \frac{8}{3} \]
\[ = -1 + \frac{4\sqrt{2}}{3} + \frac{6}{3} \]
\[ = \frac{5}{3} + \frac{4\sqrt{2}}{3} \]

### Final Answer
Thus, the value of the integral is:
\[ \int_{0}^{2} f(x) \, dx = \frac{5 + 4\sqrt{2}}{3} \]