Let the three consecutive integers be \( x \), \( x + 1 \), and \( x + 2 \).
The sum of these integers can be expressed as:
\[ x + (x + 1) + (x + 2) = 21 \]
Simplifying the left-hand side:
\[ 3x + 3 = 21 \]
Next, we subtract 3 from both sides:
\[ 3x = 21 - 3 \] \[ 3x = 18 \]
Now, we divide both sides by 3:
\[ x = 6 \]
Thus, the three consecutive integers are:
\[ x = 6, \quad x + 1 = 7, \quad x + 2 = 8 \]
Therefore, the three consecutive integers are \( 6, 7, \) and \( 8 \).