https://file.aiquickdraw.com/mathbot/file/1726808721986avc2a6ky.jpg

The integral in the image is:

\[
\int_0^1 \frac{1}{4}(1 + y) \, dy
\]

### Solution:
1. Factor out the constant \(\frac{1}{4}\):

\[
\frac{1}{4} \int_0^1 (1 + y) \, dy
\]

2. Now integrate \(1 + y\):

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

3. Apply the limits of integration (from 0 to 1):

\[
\left[ y + \frac{y^2}{2} \right]_0^1 = \left( 1 + \frac{1^2}{2} \right) - \left( 0 + \frac{0^2}{2} \right)
\]

This simplifies to:

\[
1 + \frac{1}{2} = \frac{3}{2}
\]

4. Multiply by \(\frac{1}{4}\):

\[
\frac{1}{4} \times \frac{3}{2} = \frac{3}{8}
\]

### Final Answer:
\[
\frac{3}{8}
\]

Would you like further details on this solution? Here are five related questions for further practice:

1. How would the result change if the upper limit of integration was 2 instead of 1?
2. What is the general rule for integrating polynomials?
3. How can you solve integrals with variable coefficients?
4. Can you explain why constants can be factored out of integrals?
5. How would this integral be different if it were a definite integral with limits from -1 to 1?

**Tip:** Always check if you can factor out constants to simplify integrals before proceeding with integration.

This problem involves finding the definite integral of (1/4)(1 + y) from 0 to 1. The solution includes key integral calculus concepts such as factoring constants, integrating polynomials, and applying the limits of integration. Suitable for students in Grades 11-12.

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