To solve the problem of how many integers \( x \) in the set \( \{1, 2, 3, \ldots, 100\} \) satisfy the condition that \( x^2 + x^3 \) is the square of an integer, we can follow these steps:
### Step 1: Rewrite the expression
We start with the expression \( x^2 + x^3 \). We can factor it:
\[
x^2 + x^3 = x^2(1 + x)
\]
We want \( x^2(1 + x) \) to be a perfect square.
### Step 2: Analyze the condition for a perfect square
For \( x^2(1 + x) \) to be a perfect square, both \( x^2 \) and \( 1 + x \) must be perfect squares. Since \( x^2 \) is always a perfect square for any integer \( x \), we need to ensure that \( 1 + x \) is also a perfect square.
### Step 3: Set up the equation
Let \( 1 + x = k^2 \) for some integer \( k \). Then we can express \( x \) as:
\[
x = k^2 - 1
\]
### Step 4: Determine the range for \( k \)
Since \( x \) must be in the range from 1 to 100, we can set up the inequality:
\[
1 \leq k^2 - 1 \leq 100
\]
This simplifies to:
\[
2 \leq k^2 \leq 101
\]
Taking square roots gives:
\[
\sqrt{2} \leq k \leq \sqrt{101}
\]
Calculating the approximate values, we find:
\[
1.414 \leq k \leq 10.05
\]
Thus, \( k \) can take integer values from 2 to 10.
### Step 5: Calculate possible values of \( x \)
Now we can find the corresponding values of \( x \) for each integer \( k \) from 2 to 10:
- For \( k = 2 \): \( x = 2^2 - 1 = 3 \)
- For \( k = 3 \): \( x = 3^2 - 1 = 8 \)
- For \( k = 4 \): \( x = 4^2 - 1 = 15 \)
- For \( k = 5 \): \( x = 5^2 - 1 = 24 \)
- For \( k = 6 \): \( x = 6^2 - 1 = 35 \)
- For \( k = 7 \): \( x = 7^2 - 1 = 48 \)
- For \( k = 8 \): \( x = 8^2 - 1 = 63 \)
- For \( k = 9 \): \( x = 9^2 - 1 = 80 \)
- For \( k = 10 \): \( x = 10^2 - 1 = 99 \)
### Step 6: List the valid integers
The valid integers \( x \) are:
\[
3, 8, 15, 24, 35, 48, 63, 80, 99
\]
### Step 7: Count the integers
Counting these values gives us a total of 9 integers.
### Final Answer
Thus, the number of integers \( x \) in the set \( \{1, 2, 3, \ldots, 100\} \) such that \( x^2 + x^3 \) is the square of an integer is:
\[
\boxed{9}
\]
