Let the five consecutive integers be represented as \(x, x+1, x+2, x+3, x+4\).
To find the sum of these integers, we can set up the equation:
\[ x + (x+1) + (x+2) + (x+3) + (x+4) = -35 \]
Simplifying the left side, we get:
\[ 5x + (1 + 2 + 3 + 4) = -35 \]
\[ 5x + 10 = -35 \]
To isolate \(5x\), we will subtract 10 from both sides:
\[ 5x = -35 - 10 \]
\[ 5x = -45 \]
Now, we divide both sides by 5:
\[ x = -9 \]
With \(x\) found, the five consecutive integers are:
\[ -9, -8, -7, -6, -5 \]
The greatest of these integers is:
\[ \boxed{-5} \]
To verify, we can check the sum of these integers:
\[ -9 + (-8) + (-7) + (-6) + (-5) = -35 \]
Since the sum is indeed -35, the solution is confirmed. The greatest integer is
\[ \boxed{-5} \]