First, let's calculate the approximate value of \( \frac{9}{4} \):
\[ \frac{9}{4} = 2.25 \]
Next, we want to find the square root of an integer that is close to \( 2.25 \). To do this, we can find the perfect squares around 2.25.
The square of 1 is:
\[ 1^2 = 1 \]
The square of 2 is:
\[ 2^2 = 4 \]
Since \( 2.25 \) is between \( 1 \) and \( 4 \), we look for an integer whose square root is approximately \( 2.25 \). The number whose square root is closest to \( 2.25 \) is \( 5 \), because:
\[ \sqrt{5} \approx 2.236 \]
Therefore, \( 2.25 \) is close to the square root of the integer \( 5 \).
Thus, \( \frac{9}{4} \) is close to the square root of the integer \( \boxed{5} \).