What is `int_(1)^(3) |1-x^(4)| dx` equal to ?

Options

A. `-232//5`
B. `-116//5`
C. `116//5`
D. `232//5`

Explanation

To solve the integral \( \int_{1}^{3} |1 - x^4| \, dx \), we will follow these steps:

### Step 1: Determine the points where the expression inside the absolute value changes sign.
The expression \( 1 - x^4 \) changes sign when \( 1 - x^4 = 0 \). This gives us:

\[ x^4 = 1 \implies x = 1 \]

Since we are integrating from 1 to 3, we need to evaluate the sign of \( 1 - x^4 \) in this interval.

- For \( x = 1 \): \( 1 - 1^4 = 0 \) (the expression is zero)
- For \( x = 2 \): \( 1 - 2^4 = 1 - 16 = -15 \) (the expression is negative)
- For \( x = 3 \): \( 1 - 3^4 = 1 - 81 = -80 \) (the expression is negative)

Thus, \( 1 - x^4 \) is positive at \( x = 1 \) and negative for \( x \) in the interval \( (1, 3] \).

### Step 2: Rewrite the integral without the absolute value.
Since \( 1 - x^4 \) is negative for \( x \) in \( (1, 3] \), we can rewrite the integral as:

\[ \int_{1}^{3} |1 - x^4| \, dx = \int_{1}^{3} -(1 - x^4) \, dx = \int_{1}^{3} (x^4 - 1) \, dx \]

### Step 3: Compute the integral.
Now we compute the integral:

\[ \int_{1}^{3} (x^4 - 1) \, dx = \int_{1}^{3} x^4 \, dx - \int_{1}^{3} 1 \, dx \]

Calculating each part:

1. **Integral of \( x^4 \)**:
\[ \int x^4 \, dx = \frac{x^5}{5} \Big|_{1}^{3} = \left( \frac{3^5}{5} - \frac{1^5}{5} \right) = \left( \frac{243}{5} - \frac{1}{5} \right) = \frac{242}{5} \]

2. **Integral of \( 1 \)**:
\[ \int 1 \, dx = x \Big|_{1}^{3} = 3 - 1 = 2 \]

Putting it all together:

\[ \int_{1}^{3} (x^4 - 1) \, dx = \frac{242}{5} - 2 = \frac{242}{5} - \frac{10}{5} = \frac{232}{5} \]

### Final Answer:
Thus, the value of the integral \( \int_{1}^{3} |1 - x^4| \, dx \) is:

\[ \frac{232}{5} \]