To solve the problem of finding the first of three consecutive even integers whose sum is 174, we can set up the following equation:

Let the first even integer be $x$. Then, the next two consecutive even integers will be $x + 2$ and $x + 4$.

The sum of these three integers is:

$x + (x + 2) + (x + 4) = 174$

Combining like terms, we get:

$3x + 6 = 174$

To solve for $x$, first subtract 6 from both sides:

$3x = 168$

Then, divide both sides by 3:

$x = 56$

Thus, the first number is 56.