To solve for the set $D = (A \cap C) \cup B$, we first need to define the sets $A$, $B$, and $C$ based on the set $E = \{1, 2, 3, 4, 5\}$:

1. Set $A$ (even integers in $E$): $A = \{2, 4\}$

2. Set $B$ (odd integers in $E$): $B = \{1, 3, 5\}$

3. Set $C$ (powers of 2 in $E$): The powers of 2 in the set $E$ are: $C = \{1, 2, 4\}$

Next, we calculate the intersection $A \cap C$: $A \cap C = \{2, 4\} \cap \{1, 2, 4\} = \{2, 4\}$

Now, we calculate $D$: $D = (A \cap C) \cup B = \{2, 4\} \cup \{1, 3, 5\} = \{1, 2, 3, 4, 5\}$

Thus, $D = E$.

Answer: $D = E$.

If you have any questions or need further details, feel free to ask! Here are some related questions:

1. What are the properties of set operations such as union and intersection?
2. How can we identify all the subsets of a given set?
3. What is the significance of powers of integers in set theory?
4. How would the result change if $E$ included negative integers?
5. Can you provide examples of other set operations with different sets?

Tip: When working with sets, always list the elements clearly to avoid confusion, especially when performing operations like union or intersection.