the two consecutive even integers can be written as:

2x, 2x + 2, where x is integer

2x < 2x + 2 that is 2x is the smaller

the smaller added to five times the larger gives a sum of 34:

2x + 2(2x + 2) = 34

by solving we find:

x = 5

click here to see the step by step solution of the equation:

2*5 = 10

2*5 + 2 = 12

the two consecutive even integers are 10 and 12.