Integration of 1/cos(x-a)cos(x-b)dx:

To integrate the given expression, we can use the concept of trigonometric identities and some algebraic manipulations. Let's break down the process step by step:

Step 1: Identifying the Trigonometric Identities

First, let's simplify the expression by using the trigonometric identity:

cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

Using this identity, we can rewrite the expression as:

1/cos(x-a)cos(x-b) = 1/[cos(x)cos(a) + sin(x)sin(a)][cos(x)cos(b) + sin(x)sin(b)]

Step 2: Expanding the Expression

Next, we can expand the denominator by multiplying the terms using the distributive property:

= 1/[cos(x)cos(a)cos(x)cos(b) + cos(x)cos(a)sin(x)sin(b) + sin(x)sin(a)cos(x)cos(b) + sin(x)sin(a)sin(x)sin(b)]

Step 3: Simplifying the Expression

Now, let's simplify the expanded expression by combining like terms:

= 1/[cos(x)cos(x)cos(a)cos(b) + sin(x)sin(x)sin(a)sin(b) + cos(x)sin(x)cos(a)sin(b) + sin(x)cos(x)sin(a)cos(b)]

= 1/[cos^2(x)cos(a)cos(b) + sin^2(x)sin(a)sin(b) + sin(x)cos(x)[cos(a)sin(b) + sin(a)cos(b)]]

Step 4: Applying the Trigonometric Identity

Now, we can use the trigonometric identity:

cos^2(x) + sin^2(x) = 1

to simplify the expression further:

= 1/[cos(a)cos(b) + sin(a)sin(b) + sin(x)cos(x)[cos(a)sin(b) + sin(a)cos(b)]]

Step 5: Integrating the Expression

Finally, we can integrate the expression by treating the terms as separate components:

∫[1/cos(x-a)cos(x-b)]dx = ∫[1/[cos(a)cos(b) + sin(a)sin(b) + sin(x)cos(x)[cos(a)sin(b) + sin(a)cos(b)]]dx

= ∫[1/(cos(a)cos(b) + sin(a)sin(b))]dx + ∫[sin(x)cos(x)[cos(a)sin(b) + sin(a)cos(b)]]dx

The first integral can be easily evaluated as:

∫[1/(cos(a)cos(b) + sin(a)sin(b))]dx = (1/(cos(a)cos(b) + sin(a)sin(b)))x + C1

where C1 is the constant of integration.

The second integral can be simplified using the substitution u = sin(x)cos(x), which leads to du = cos^2(x) - sin^2(x) dx:

∫[sin(x)cos(x)[cos(a)sin(b) + sin(a)cos(b)]]dx = ∫