Divide it by Sin (b-a) and multiply by sin[(x-a) _(x-b) ] and expand the numerator

Integration of 1/ sin(x-a)sin(x-b) dx

- Strategy:
- We can use the product-to-sum identities for sine functions to simplify the integral.
- The product-to-sum identities are: sin(A)sin(B) = 1/2[cos(A-B) - cos(A+B)].

- Step 1: Rewrite the integral
- Rewrite the integrand as: 1/2[sin(a-b)cos(2x-(a+b)) - sin(a+b)cos(2x-(a-b))] / sin(x-a)sin(x-b).

- Step 2: Simplify the integral
- Use partial fraction decomposition to split the integrand into simpler terms.
- Apply the product-to-sum identities to simplify the integrand further.

- Step 3: Integrate the simplified terms
- Integrate each term obtained from the simplification step.
- Use the standard integration techniques for trigonometric functions.

- Step 4: Combine the results
- Combine the results of the integrations to obtain the final expression.

By following these steps and manipulating the integrand using trigonometric identities, we can evaluate the integral of 1/ sin(x-a)sin(x-b) dx.