`int|x|^3 dx` is equal to

Options

A. `(x^4)/(4)`
B. `- (x^4)/(4)`
C. `(|x|^4)/(4)`
D. none of these

Answer

To solve the integral \(\int |x|^3 \, dx\), we need to consider the absolute value function, which behaves differently depending on whether \(x\) is positive or negative.

### Step-by-Step Solution:

1. **Break Down the Absolute Value**:
The absolute value function \(|x|\) can be defined as:
\[ |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases} \]
Therefore, \(|x|^3\) can be expressed as:
\[ |x|^3 = \begin{cases} x^3 & \text{if } x \geq 0 \\ (-x)^3 = -x^3 & \text{if } x < 0 \end{cases} \]

2. **Set Up the Integral for Each Case**:
We will evaluate the integral in two cases based on the definition of \(|x|^3\):
- Case 1: \(x \geq 0\)
\[ \int |x|^3 \, dx = \int x^3 \, dx \]
- Case 2: \(x < 0\)
\[ \int |x|^3 \, dx = \int -x^3 \, dx \]

3. **Integrate for Case 1** (\(x \geq 0\)):
\[ \int x^3 \, dx = \frac{x^4}{4} + C_1 \]

4. **Integrate for Case 2** (\(x < 0\)):
\[ \int -x^3 \, dx = -\left(\frac{x^4}{4}\right) + C_2 = -\frac{x^4}{4} + C_2 \]

5. **Combine the Results**:
Now we can express the integral \(\int |x|^3 \, dx\) as:
\[ \int |x|^3 \, dx = \begin{cases} \frac{x^4}{4} + C_1 & \text{if } x \geq 0 \\ -\frac{x^4}{4} + C_2 & \text{if } x < 0 \end{cases} \]

6. **Final Expression**:
Since the constants of integration \(C_1\) and \(C_2\) can be combined into a single constant \(C\), we can write:
\[ \int |x|^3 \, dx = \begin{cases} \frac{x^4}{4} + C & \text{if } x \geq 0 \\ -\frac{x^4}{4} + C & \text{if } x < 0 \end{cases} \]