Evaluate the following integrals .
`int_(a)^(b)x^(3)dx`

Explanation

To evaluate the integral \( I = \int_{a}^{b} x^3 \, dx \), we can follow these steps:

### Step 1: Identify the integral
We need to evaluate the definite integral of \( x^3 \) from \( a \) to \( b \).

### Step 2: Use the power rule for integration
The power rule for integration states that:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
For our integral, \( n = 3 \), so we apply the formula:
\[ \int x^3 \, dx = \frac{x^{3+1}}{3+1} = \frac{x^4}{4} \]

### Step 3: Evaluate the definite integral
Now, we need to evaluate the integral from \( a \) to \( b \):
\[ I = \left[ \frac{x^4}{4} \right]_{a}^{b} \]
This means we will substitute \( b \) and \( a \) into the expression \( \frac{x^4}{4} \).

### Step 4: Substitute the limits
Substituting the upper limit \( b \):
\[ \frac{b^4}{4} \]
Substituting the lower limit \( a \):
\[ \frac{a^4}{4} \]

### Step 5: Calculate the difference
Now, we find the difference between the two results:
\[ I = \frac{b^4}{4} - \frac{a^4}{4} \]
This can be simplified to:
\[ I = \frac{1}{4} (b^4 - a^4) \]

### Final Result
Thus, the value of the integral is:
\[ I = \frac{1}{4} (b^4 - a^4) \]

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