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To solve the integral \( \int y^2 (y + 2)^{\frac{1}{4}} \, dy \), we'll use the method of substitution. Here's the step-by-step solution:

1. **Substitution**: Let \( u = y + 2 \).
   - Then \( du = dy \) and \( y = u - 2 \).
   - When \( y = u - 2 \), \( y^2 = (u - 2)^2 \).

2. **Rewrite the integral**:
   \[
   \int y^2 (y + 2)^{\frac{1}{4}} \, dy = \int (u - 2)^2 u^{\frac{1}{4}} \, du
   \]

3. **Expand the integrand**:
   \[
   (u - 2)^2 = u^2 - 4u + 4
   \]
   So,
   \[
   \int (u^2 - 4u + 4) u^{\frac{1}{4}} \, du = \int (u^2 u^{\frac{1}{4}} - 4u u^{\frac{1}{4}} + 4 u^{\frac{1}{4}}) \, du
   \]

4. **Simplify the integrand**:
   \[
   \int (u^{2 + \frac{1}{4}} - 4u^{1 + \frac{1}{4}} + 4u^{\frac{1}{4}}) \, du = \int (u^{\frac{9}{4}} - 4u^{\frac{5}{4}} + 4u^{\frac{1}{4}}) \, du
   \]

5. **Integrate term by term**:
   \[
   \int u^{\frac{9}{4}} \, du - 4 \int u^{\frac{5}{4}} \, du + 4 \int u^{\frac{1}{4}} \, du
   \]

   Using the power rule \( \int u^n \, du = \frac{u^{n+1}}{n+1} \), we get:
   \[
   \int u^{\frac{9}{4}} \, du = \frac{u^{\frac{9}{4} + 1}}{\frac{9}{4} + 1} = \frac{u^{\frac{13}{4}}}{\frac{13}{4}} = \frac{4}{13} u^{\frac{13}{4}}
   \]
   \[
   - 4 \int u^{\frac{5}{4}} \, du = -4 \cdot \frac{u^{\frac{5}{4} + 1}}{\frac{5}{4} + 1} = -4 \cdot \frac{u^{\frac{9}{4}}}{\frac{9}{4}} = -\frac{16}{9} u^{\frac{9}{4}}
   \]
   \[
   + 4 \int u^{\frac{1}{4}} \, du = 4 \cdot \frac{u^{\frac{1}{4} + 1}}{\frac{1}{4} + 1} = 4 \cdot \frac{u^{\frac{5}{4}}}{\frac{5}{4}} = \frac{16}{5} u^{\frac{5}{4}}
   \]

6. **Combine the results**:
   \[
   \int (u^{\frac{9}{4}} - 4u^{\frac{5}{4}} + 4u^{\frac{1}{4}}) \, du = \frac{4}{13} u^{\frac{13}{4}} - \frac{16}{9} u^{\frac{9}{4}} + \frac{16}{5} u^{\frac{5}{4}} + C
   \]

7. **Substitute back \( u = y + 2 \)**:
   \[
   \frac{4}{13} (y + 2)^{\frac{13}{4}} - \frac{16}{9} (y + 2)^{\frac{9}{4}} + \frac{16}{5} (y + 2)^{\frac{5}{4}} + C
   \]

So the solution to the integral is:
\[
\int y^2 (y + 2)^{\frac{1}{4}} \, dy = \frac{4}{13} (y + 2)^{\frac{13}{4}} - \frac{16}{9} (y + 2)^{\frac{9}{4}} + \frac{16}{5} (y + 2)^{\frac{5}{4}} + C
\]

Would you like further details or have any questions?

Here are some related questions you might find helpful:
1. How do you choose the right substitution for integrals?
2. What is the power rule for integration and when can it be applied?
3. Can you provide an example of integration by parts?
4. How do you handle integrals involving trigonometric functions?
5. What are some common techniques for solving definite integrals?
6. How do you determine the constant of integration?
7. What are improper integrals and how are they evaluated?
8. Can you explain the Fundamental Theorem of Calculus?

**Tip**: Always double-check your substitutions and limits when solving integrals to avoid errors in your final answer.

Learn how to solve the integral \( \int y^2 (y + 2)^{\frac{1}{4}} \, dy \) using substitution method with a detailed step-by-step solution. This problem covers integration techniques, including substitution and power rule, suitable for advanced high school students.

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