If `int|x| dx = kx |x| + C, x cancel(=) 0`, then the value of k is

Options

A. 2
B. `-2`
C. `-(1)/(2)`
D. `(1)/(2)`

Answer

To solve the problem of finding the value of \( k \) in the equation

\[ \int |x| \, dx = kx |x| + C, \quad x \neq 0, \]

we will break down the integration of \( |x| \) into two cases based on the definition of the absolute value function.

### Step 1: Define the cases for \( |x| \)
The absolute value function \( |x| \) can be defined as:
- \( |x| = x \) when \( x \geq 0 \)
- \( |x| = -x \) when \( x < 0 \)

### Step 2: Integrate for \( x \geq 0 \)
For \( x \geq 0 \):
\[ \int |x| \, dx = \int x \, dx = \frac{x^2}{2} + C_1 \]

### Step 3: Integrate for \( x < 0 \)
For \( x < 0 \):
\[ \int |x| \, dx = \int -x \, dx = -\frac{x^2}{2} + C_2 \]

### Step 4: Combine the results
We can express the results of the integration in a piecewise manner:
\[ \int |x| \, dx = \begin{cases} \frac{x^2}{2} + C_1 & \text{if } x \geq 0 \\ -\frac{x^2}{2} + C_2 & \text{if } x < 0 \end{cases} \]

### Step 5: Express the result in terms of \( |x| \)
We can rewrite both cases in terms of \( |x| \):
\[ \int |x| \, dx = |x| \cdot \frac{x}{2} + C \]
This is because:
- For \( x \geq 0 \), \( |x| = x \), so \( |x| \cdot \frac{x}{2} = \frac{x^2}{2} \).
- For \( x < 0 \), \( |x| = -x \), so \( |x| \cdot \frac{x}{2} = -\frac{x^2}{2} \).

### Step 6: Compare with the given equation
Now, we compare this result with the given equation:
\[ \int |x| \, dx = kx |x| + C \]
From our derived expression, we see that:
\[ kx |x| = |x| \cdot \frac{x}{2} \]
This implies:
\[ k = \frac{1}{2} \]

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{\frac{1}{2}} \]