To find the integer that \( \frac{9}{4} \) is close to the square root of, we first need to calculate \( \frac{9}{4} \):
\[ \frac{9}{4} = 2.25 \]
Next, we need to find the integer \( n \) such that \( \sqrt{n} \) is close to \( 2.25 \). To do this, we will square \( 2.25 \):
\[ (2.25)^2 = 5.0625 \]
Now, we can round \( 5.0625 \) to the nearest integer, which is \( 5 \).
Therefore, \( \frac{9}{4} \) is close to the square root of the integer \( \boxed{5} \).