Evaluate the following integrals:
`int_(-1)^(1)x^(3)|x|dx`

Answer

To evaluate the integral \( \int_{-1}^{1} x^{3} |x| \, dx \), we will break it down into two parts based on the definition of the absolute value function.

### Step 1: Understand the Absolute Value Function
The absolute value function \( |x| \) can be defined as:
- \( |x| = x \) when \( x \geq 0 \)
- \( |x| = -x \) when \( x < 0 \)

### Step 2: Break the Integral into Two Parts
Since the integral is from \(-1\) to \(1\), we will split it at \(0\):
\[ \int_{-1}^{1} x^{3} |x| \, dx = \int_{-1}^{0} x^{3} |x| \, dx + \int_{0}^{1} x^{3} |x| \, dx \]

### Step 3: Evaluate Each Integral Separately
1. **For the interval \([-1, 0]\)**:
Here, \( |x| = -x \), so:
\[ \int_{-1}^{0} x^{3} |x| \, dx = \int_{-1}^{0} x^{3} (-x) \, dx = \int_{-1}^{0} -x^{4} \, dx \]

2. **For the interval \([0, 1]\)**:
Here, \( |x| = x \), so:
\[ \int_{0}^{1} x^{3} |x| \, dx = \int_{0}^{1} x^{3} x \, dx = \int_{0}^{1} x^{4} \, dx \]

### Step 4: Combine the Integrals
Now we can combine the two integrals:
\[ \int_{-1}^{1} x^{3} |x| \, dx = \int_{-1}^{0} -x^{4} \, dx + \int_{0}^{1} x^{4} \, dx \]

### Step 5: Evaluate Each Integral
1. **Evaluate \( \int_{-1}^{0} -x^{4} \, dx \)**:
\[ \int -x^{4} \, dx = -\frac{x^{5}}{5} \Bigg|_{-1}^{0} = -\left(0 - \left(-\frac{(-1)^{5}}{5}\right)\right) = -\left(0 + \frac{1}{5}\right) = -\frac{1}{5} \]

2. **Evaluate \( \int_{0}^{1} x^{4} \, dx \)**:
\[ \int x^{4} \, dx = \frac{x^{5}}{5} \Bigg|_{0}^{1} = \frac{1^{5}}{5} - \frac{0^{5}}{5} = \frac{1}{5} - 0 = \frac{1}{5} \]

### Step 6: Combine the Results
Now we can combine the results of the two integrals:
\[ \int_{-1}^{1} x^{3} |x| \, dx = -\frac{1}{5} + \frac{1}{5} = 0 \]

### Final Answer
Thus, the value of the integral \( \int_{-1}^{1} x^{3} |x| \, dx \) is:
\[ \boxed{0} \]