The value of `int_(0)^(1)x|x-1/2|dx` is

Options

A. `1/3`
B. `1/4`
C. `1/8`
D. None of these

Answer

To solve the integral \( \int_{0}^{1} x |x - \frac{1}{2}| \, dx \), we first need to analyze the expression \( |x - \frac{1}{2}| \) within the limits of integration.

### Step 1: Determine the behavior of \( |x - \frac{1}{2}| \)

The expression \( |x - \frac{1}{2}| \) can be broken down based on the value of \( x \):
- For \( x < \frac{1}{2} \), \( |x - \frac{1}{2}| = \frac{1}{2} - x \)
- For \( x \geq \frac{1}{2} \), \( |x - \frac{1}{2}| = x - \frac{1}{2} \)

Since our integral runs from 0 to 1, we can split the integral at \( x = \frac{1}{2} \):

\[ \int_{0}^{1} x |x - \frac{1}{2}| \, dx = \int_{0}^{\frac{1}{2}} x \left( \frac{1}{2} - x \right) \, dx + \int_{\frac{1}{2}}^{1} x \left( x - \frac{1}{2} \right) \, dx \]

### Step 2: Evaluate the first integral \( \int_{0}^{\frac{1}{2}} x \left( \frac{1}{2} - x \right) \, dx \)

We can simplify the first integral:

\[ \int_{0}^{\frac{1}{2}} x \left( \frac{1}{2} - x \right) \, dx = \int_{0}^{\frac{1}{2}} \left( \frac{1}{2}x - x^2 \right) \, dx \]

Now, we can integrate term by term:

\[ = \left[ \frac{1}{4}x^2 - \frac{1}{3}x^3 \right]_{0}^{\frac{1}{2}} \]

Calculating the limits:

\[ = \left( \frac{1}{4} \cdot \left( \frac{1}{2} \right)^2 - \frac{1}{3} \cdot \left( \frac{1}{2} \right)^3 \right) - \left( 0 - 0 \right) \]
\[ = \left( \frac{1}{4} \cdot \frac{1}{4} - \frac{1}{3} \cdot \frac{1}{8} \right) \]
\[ = \frac{1}{16} - \frac{1}{24} \]

Finding a common denominator (48):

\[ = \frac{3}{48} - \frac{2}{48} = \frac{1}{48} \]

### Step 3: Evaluate the second integral \( \int_{\frac{1}{2}}^{1} x \left( x - \frac{1}{2} \right) \, dx \)

Now, we evaluate the second integral:

\[ \int_{\frac{1}{2}}^{1} x \left( x - \frac{1}{2} \right) \, dx = \int_{\frac{1}{2}}^{1} \left( x^2 - \frac{1}{2}x \right) \, dx \]

Integrating term by term:

\[ = \left[ \frac{1}{3}x^3 - \frac{1}{4}x^2 \right]_{\frac{1}{2}}^{1} \]

Calculating the limits:

\[ = \left( \frac{1}{3}(1)^3 - \frac{1}{4}(1)^2 \right) - \left( \frac{1}{3}\left( \frac{1}{2} \right)^3 - \frac{1}{4}\left( \frac{1}{2} \right)^2 \right) \]
\[ = \left( \frac{1}{3} - \frac{1}{4} \right) - \left( \frac{1}{3} \cdot \frac{1}{8} - \frac{1}{4} \cdot \frac{1}{4} \right) \]
\[ = \left( \frac{4}{12} - \frac{3}{12} \right) - \left( \frac{1}{24} - \frac{1}{16} \right) \]
Finding a common denominator (48):

\[ = \frac{1}{12} - \left( \frac{2}{48} - \frac{3}{48} \right) = \frac{1}{12} + \frac{1}{48} \]

Finding a common denominator (48):

\[ = \frac{4}{48} + \frac{1}{48} = \frac{5}{48} \]

### Step 4: Combine the results

Now we combine both integrals:

\[ \int_{0}^{1} x |x - \frac{1}{2}| \, dx = \frac{1}{48} + \frac{5}{48} = \frac{6}{48} = \frac{1}{8} \]

### Final Answer

Thus, the value of the integral \( \int_{0}^{1} x |x - \frac{1}{2}| \, dx \) is:

\[ \boxed{\frac{1}{8}} \]