`int x^(8) " dx "`

Answer

To solve the integral \( \int x^8 \, dx \), we will use the power rule of integration. Here’s a step-by-step solution:

### Step 1: Write down the integral
We start with the integral we want to evaluate:
\[ I = \int x^8 \, dx \]

### Step 2: Apply the power rule of integration
According to the power rule of integration, the integral of \( x^n \) is given by:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
where \( C \) is the constant of integration.

### Step 3: Identify \( n \)
In our case, \( n = 8 \).

### Step 4: Substitute \( n \) into the formula
Now we substitute \( n = 8 \) into the power rule:
\[ I = \frac{x^{8+1}}{8+1} + C \]

### Step 5: Simplify the expression
This simplifies to:
\[ I = \frac{x^9}{9} + C \]

### Final Answer
Thus, the integral \( \int x^8 \, dx \) is:
\[ \int x^8 \, dx = \frac{x^9}{9} + C \]

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