If n is a positive odd integer, then `int |x^n| dx=`

Options

A. `|(x^(n+1))/(n+1)|+C`
B. `(x^(n+1))/(n+1)+C`
C. `(|x|^(n)x)/(n+1)+C`
D. None of these

Answer

To solve the integral \( \int |x^n| \, dx \) where \( n \) is a positive odd integer, we can follow these steps:

### Step 1: Understand the Absolute Value
Since \( n \) is a positive odd integer, the expression \( x^n \) will be negative when \( x < 0 \) and positive when \( x > 0 \). Therefore, we can express the absolute value as follows:
\[ |x^n| = \begin{cases} -x^n & \text{if } x < 0 \\ x^n & \text{if } x \geq 0 \end{cases} \]

### Step 2: Set Up the Integral
We can break the integral into two cases based on the sign of \( x \):
\[ \int |x^n| \, dx = \begin{cases} \int -x^n \, dx & \text{if } x < 0 \\ \int x^n \, dx & \text{if } x \geq 0 \end{cases} \]

### Step 3: Integrate Each Case
1. **For \( x < 0 \)**:
\[ \int -x^n \, dx = -\frac{x^{n+1}}{n+1} + C_1 \]

2. **For \( x \geq 0 \)**:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C_2 \]

### Step 4: Combine the Results
Now we can combine the results from both cases:
\[ \int |x^n| \, dx = \begin{cases} -\frac{x^{n+1}}{n+1} + C_1 & \text{if } x < 0 \\ \frac{x^{n+1}}{n+1} + C_2 & \text{if } x \geq 0 \end{cases} \]

### Step 5: Express in Terms of Absolute Value
To express this in a single formula, we can use the property of absolute values:
\[ \int |x^n| \, dx = \frac{x^{n+1}}{n+1} \text{sgn}(x) + C \]
where \( \text{sgn}(x) \) is the sign function, which is -1 for \( x < 0 \) and +1 for \( x > 0 \).

### Final Answer
Thus, the final result can be summarized as:
\[ \int |x^n| \, dx = \frac{|x|^{n+1}}{n+1} + C \]