Math Problem Statement

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

Solution

The integral provided is:

8xdx\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 xx can be written as a fractional exponent: 8xdx=8x1/2dx\int 8 \sqrt{x} \, dx = \int 8 x^{1/2} \, dx

2. Integrate:

To integrate 8x1/28x^{1/2}, we apply the power rule for integration, which states:

xndx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C

For n=12n = \frac{1}{2}, this gives:

8x1/2dx=8x12+112+1+C=8x3/23/2+C\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:

=823x3/2+C=163x3/2+C= 8 \cdot \frac{2}{3} x^{3/2} + C = \frac{16}{3} x^{3/2} + C

3. Simplify:

The simplified

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Math Problem Analysis

Mathematical Concepts

Integration
Exponents

Formulas

Power Rule for Integration: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)

Theorems

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Suitable Grade Level

Grades 11-12