**Integration of (cosx - sinx)^2**

To integrate the expression (cosx - sinx)^2, we can expand it and then integrate each term separately. Let's break it down step by step:

**Step 1: Expand the expression**

First, we expand the square of the binomial (cosx - sinx)^2 using the formula (a - b)^2 = a^2 - 2ab + b^2:

(cosx - sinx)^2 = cos^2(x) - 2cos(x)sin(x) + sin^2(x)

**Step 2: Simplify the expression**

Next, we simplify the expression by using the trigonometric identity cos^2(x) + sin^2(x) = 1:

(cosx - sinx)^2 = 1 - 2cos(x)sin(x)

**Step 3: Integrate each term**

Now, we integrate each term separately. The integral of 1 with respect to x is simply x:

∫1 dx = x + C (where C is the constant of integration)

To integrate the term -2cos(x)sin(x), we can use the substitution method. Let's substitute u = sin(x) and find du:

u = sin(x)
du = cos(x) dx

Now, we can rewrite the integral in terms of u:

∫-2cos(x)sin(x) dx = -2∫u du

Integrating -2∫u du gives us -u^2:

-2∫u du = -2(u^2) = -2sin^2(x)

**Step 4: Combine the integrals and simplify**

Now, let's combine the two integrals:

∫(cosx - sinx)^2 dx = x - 2sin^2(x) + C

where C is the constant of integration.

Thus, the integration of (cosx - sinx)^2 is x - 2sin^2(x) + C.

Note: In the above steps, we have assumed that the integration is with respect to x. Make sure to specify the variable of integration if it is different from x.