https://r2.erweima.ai/mathbot/image/1718441409608b1yu93gj.jpeg

To solve the integral 

\[ \int y^2 (y + 2)^{\frac{1}{4}} \, dy \]

we can use a substitution method. Let's define a substitution to simplify the integral.

Let \( u = y + 2 \). Then, \( du = dy \) and \( y = u - 2 \).

Substitute these into the integral:

\[ \int y^2 (y + 2)^{\frac{1}{4}} \, dy = \int (u - 2)^2 u^{\frac{1}{4}} \, du \]

Now, expand the integrand:

\[ (u - 2)^2 = u^2 - 4u + 4 \]

So the integral becomes:

\[ \int (u^2 - 4u + 4) u^{\frac{1}{4}} \, du \]

Distribute \( u^{\frac{1}{4}} \):

\[ \int (u^2 u^{\frac{1}{4}} - 4u u^{\frac{1}{4}} + 4 u^{\frac{1}{4}}) \, du = \int (u^{2 + \frac{1}{4}} - 4u^{1 + \frac{1}{4}} + 4u^{\frac{1}{4}}) \, du \]

Simplify the exponents:

\[ \int (u^{\frac{9}{4}} - 4u^{\frac{5}{4}} + 4u^{\frac{1}{4}}) \, du \]

Now integrate term by term:

\[ \int u^{\frac{9}{4}} \, du = \frac{u^{\frac{9}{4} + 1}}{\frac{9}{4} + 1} = \frac{u^{\frac{13}{4}}}{\frac{13}{4}} = \frac{4}{13} u^{\frac{13}{4}} \]

\[ \int -4u^{\frac{5}{4}} \, du = -4 \cdot \frac{u^{\frac{5}{4} + 1}}{\frac{5}{4} + 1} = -4 \cdot \frac{u^{\frac{9}{4}}}{\frac{9}{4}} = -4 \cdot \frac{4}{9} u^{\frac{9}{4}} = -\frac{16}{9} u^{\frac{9}{4}} \]

\[ \int 4u^{\frac{1}{4}} \, du = 4 \cdot \frac{u^{\frac{1}{4} + 1}}{\frac{1}{4} + 1} = 4 \cdot \frac{u^{\frac{5}{4}}}{\frac{5}{4}} = 4 \cdot \frac{4}{5} u^{\frac{5}{4}} = \frac{16}{5} u^{\frac{5}{4}} \]

Combine all these terms:

\[ \frac{4}{13} u^{\frac{13}{4}} - \frac{16}{9} u^{\frac{9}{4}} + \frac{16}{5} u^{\frac{5}{4}} + C \]

Substitute back \( u = y + 2 \):

\[ \frac{4}{13} (y + 2)^{\frac{13}{4}} - \frac{16}{9} (y + 2)^{\frac{9}{4}} + \frac{16}{5} (y + 2)^{\frac{5}{4}} + C \]

This is the evaluated integral.

Would you like more details or have any questions?

Here are 5 related questions you might be interested in:

1. How do you solve integrals using integration by parts?
2. What are common techniques for solving definite integrals?
3. How do you approach integrals with trigonometric functions?
4. What is the method for integrating rational functions?
5. How do you use partial fraction decomposition in integration?

**Tip:** When solving integrals, substitution is a powerful technique to simplify the integral into a more manageable form. Always look for potential substitutions that can make the integrand simpler.

Learn how to solve the integral ∫ y^2 (y + 2)^(1/4) dy step-by-step using the substitution method. This problem covers advanced integration techniques and is suitable for advanced high school or undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation