Let the first integer be \( x \). Since we are looking for two consecutive integers, the second integer can be expressed as \( x + 1 \).
According to the problem, the sum of these two integers is 37. We can write the equation as follows:
\[ x + (x + 1) = 37 \]
Simplifying the equation:
\[ 2x + 1 = 37 \]
Next, we subtract 1 from both sides:
\[ 2x = 36 \]
Now, divide both sides by 2:
\[ x = 18 \]
Thus, the first integer is \( 18 \), and the second integer, being consecutive, is:
\[ x + 1 = 19 \]
Therefore, the two consecutive integers are 18 and 19.