**Solution:**

Let's assume the smaller integer as x and the larger integer as y.

Given:
The difference of two positive integers is 3.

So, we can write the equation: y - x = 3 ...(1)

The sum of the squares of the integers is 89.

So, we can write the equation: x^2 + y^2 = 89 ...(2)

We have two equations (1) and (2) with two unknowns (x and y). So, we can solve these equations simultaneously to find the values of x and y.

**Solving the Equations:**

To solve these equations, we can use the substitution method.

From equation (1), we can write y = x + 3.

Substituting this value of y in equation (2), we get:

x^2 + (x + 3)^2 = 89

Expanding and simplifying the equation:

x^2 + x^2 + 6x + 9 = 89

Combining like terms:

2x^2 + 6x + 9 - 89 = 0

2x^2 + 6x - 80 = 0

Dividing the equation by 2, we get:

x^2 + 3x - 40 = 0

**Using the Quadratic Formula:**

The quadratic equation is of the form ax^2 + bx + c = 0, where a = 1, b = 3, and c = -40.

Using the quadratic formula, x = (-b ± √(b^2 - 4ac))/(2a), we can find the values of x.

Substituting the values, we get:

x = (-3 ± √(3^2 - 4(1)(-40)))/(2(1))

x = (-3 ± √(9 + 160))/(2)

x = (-3 ± √(169))/(2)

x = (-3 ± 13)/(2)

So, we have two possible values for x:

x1 = (-3 + 13)/2 = 10/2 = 5

x2 = (-3 - 13)/2 = -16/2 = -8

Since the given question specifies positive integers, we discard the negative value.

Therefore, the smaller integer x = 5.

**Finding the Larger Integer:**

From equation (1), y = x + 3, substituting the value of x, we get:

y = 5 + 3 = 8

Therefore, the larger integer y = 8.

**Conclusion:**

The smaller integer is 5 and the larger integer is 8.

Let y > x
y - x = 3 ---(1)
y^2 + x^2 = 89 ---(2)

Squaring eq (1)
y^2 + x^2 - 2yx = 9
Substituting from eq (2)

89 - 2yx = 9
2yx = 80
yx = 40

Substituting y = 3 + x

(3 + x)x = 40
x^2 + 3x - 40 = 0

x = 5
y = 8