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To complete this table, let’s go through each step for the given integral:

### Original Integral
\[
\int \sqrt[3]{x} \, dx
\]

### Step 1: Rewrite
The cube root of \(x\) can be expressed as a fractional exponent:
\[
\int x^{\frac{1}{3}} \, dx
\]

### Step 2: Integrate
To integrate \( x^{\frac{1}{3}} \), apply the power rule for integration, which states:
\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{for } n \neq -1
\]
In this case, \( n = \frac{1}{3} \), so:
\[
\int x^{\frac{1}{3}} \, dx = \frac{x^{\frac{1}{3} + 1}}{\frac{1}{3} + 1} + C = \frac{x^{\frac{4}{3}}}{\frac{4}{3}} + C
\]

### Step 3: Simplify
To simplify, divide by \( \frac{4}{3} \) (which is equivalent to multiplying by \( \frac{3}{4} \)):
\[
= \frac{3}{4} x^{\frac{4}{3}} + C
\]

### Completed Table
Here’s how the completed table would look:

| Original Integral         | Rewrite               | Integrate                            | Simplify                      |
|---------------------------|-----------------------|--------------------------------------|--------------------------------|
| \( \int \sqrt[3]{x} \, dx \) | \( \int x^{\frac{1}{3}} \, dx \) | \( \frac{x^{\frac{4}{3}}}{\frac{4}{3}} + C \) | \( \frac{3}{4} x^{\frac{4}{3}} + C \) |

Would you like further details on any of the steps?

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Here are some related questions to deepen your understanding:
1. How do you integrate functions with fractional exponents?
2. What is the power rule for integration, and when does it apply?
3. How can you rewrite radical expressions as expressions with exponents?
4. Why is a constant of integration (\(C\)) added in indefinite integrals?
5. How can you simplify expressions involving division by a fraction?

**Tip:** When integrating functions with roots, rewrite the root as a fractional exponent for easier handling.

Complete the table. (Use C for the constant of integration.)

Learn how to complete an integration table by rewriting roots as fractional exponents and applying the power rule for integration. This example focuses on integrating x^(1/3) with a detailed solution and simplification steps. Perfect for calculus students looking to strengthen their integration skills.

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