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The integral provided is:

\[
\int 8 \sqrt{x} \, dx
\]

Let's complete each section of the table step by step.

### 1. **Rewrite the Original Integral**:
The square root of \(x\) can be written as a fractional exponent:
\[
\int 8 \sqrt{x} \, dx = \int 8 x^{1/2} \, dx
\]

### 2. **Integrate**:
To integrate \(8x^{1/2}\), we apply the power rule for integration, which states:

\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C
\]

For \(n = \frac{1}{2}\), this gives:

\[
\int 8 x^{1/2} \, dx = 8 \cdot \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} + C = 8 \cdot \frac{x^{3/2}}{3/2} + C
\]

Simplify the fraction:

\[
= 8 \cdot \frac{2}{3} x^{3/2} + C = \frac{16}{3} x^{3/2} + C
\]

### 3. **Simplify**:
The simplified

Complete the table with the integral \( \int 8 \sqrt{x} \, dx \). Rewrite the integral, integrate it, and simplify.

Learn how to solve the integral \( \int 8 \sqrt{x} \, dx \) with a detailed step-by-step solution. This problem covers key calculus concepts, including rewriting the integral with exponents and applying the power rule for integration. Suitable for students in Grades 11-12.

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