x + y = 6

y = 6 - x

plug y = 6 - x into xy

x(6 - x) = 6x - x^2

we look for the maximum value of the function

f(x) = 6x - x^2

f(x) is parabolic function with negative quotient in front of x^2 that is -1<0 thus it maximum in the vertex

the maximum is calculated by the formula c - b^2/(4a) where a = -1, b = 6 and c = 0

0^2 - 6^2/(4*(-1)) = 36/4 = 9

lets check if x and y are integers

from 6x - x^2 = 9 we find the solution

x = 3 is integer

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y = 6 - x = 6 - 3 = 3 is integer

if we draw the graph we can also see that the function has a maximum equal to 9

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the largest integral value of xy is 9.