What is `int_1^3 |1-x^4| dx` equal to

Options

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

Answer

To solve the integral \( \int_1^3 |1 - x^4| \, dx \), we will follow these steps:

### Step 1: Analyze the function inside the modulus
We need to determine when \( 1 - x^4 \) is positive or negative in the interval from 1 to 3.

- At \( x = 1 \):
\[ 1 - 1^4 = 1 - 1 = 0 \]
- At \( x = 3 \):
\[ 1 - 3^4 = 1 - 81 = -80 \]

Since \( 1 - x^4 \) changes from 0 at \( x = 1 \) to negative values as \( x \) increases to 3, we can conclude that \( 1 - x^4 \) is non-positive in the interval \( [1, 3] \).

### Step 2: Rewrite the integral without the modulus
Since \( 1 - x^4 \) is negative in the interval \( [1, 3] \), 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 will compute the integral:
\[ \int_1^3 (x^4 - 1) \, dx = \int_1^3 x^4 \, dx - \int_1^3 1 \, dx \]

Calculating each part separately:
1. For \( \int_1^3 x^4 \, dx \):
\[ \int x^4 \, dx = \frac{x^5}{5} \quad \text{(antiderivative)} \]
Evaluating from 1 to 3:
\[ \left[ \frac{x^5}{5} \right]_1^3 = \frac{3^5}{5} - \frac{1^5}{5} = \frac{243}{5} - \frac{1}{5} = \frac{242}{5} \]

2. For \( \int_1^3 1 \, dx \):
\[ \int 1 \, dx = x \quad \text{(antiderivative)} \]
Evaluating from 1 to 3:
\[ \left[ x \right]_1^3 = 3 - 1 = 2 \]

### Step 4: Combine the results
Now we combine the results of the two integrals:
\[ \int_1^3 (x^4 - 1) \, dx = \frac{242}{5} - 2 = \frac{242}{5} - \frac{10}{5} = \frac{232}{5} \]

### Final Answer
Thus, the value of the integral \( \int_1^3 |1 - x^4| \, dx \) is:
\[ \frac{232}{5} \]

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