Integration of (cos5x cos4x)/(1-2cos3x)

To integrate the given expression, we can use the trigonometric identity:

cos A cos B = 1/2 [cos(A + B) + cos(A - B)]

Let's break down the expression and simplify it using this identity.

1. Simplifying the numerator:
cos5x cos4x = 1/2 [cos(5x + 4x) + cos(5x - 4x)]
= 1/2 [cos(9x) + cos(x)]

2. Simplifying the denominator:
1 - 2cos3x

3. Rewriting the expression:
(cos5x cos4x)/(1 - 2cos3x) = [1/2 (cos(9x) + cos(x))]/(1 - 2cos3x)

Now, let's proceed with the integration using a substitution method.

4. Substitute u = cos(x):
To simplify the integration, we will substitute u = cos(x). Therefore, du = -sin(x) dx.

5. Rewriting the expression with the substitution:
[1/2 (cos(9x) + cos(x))]/(1 - 2cos3x) = [1/2 (cos(9x) + u)]/(1 - 2u^3) du

6. Integrating the expression:
∫ [1/2 (cos(9x) + u)]/(1 - 2u^3) du

7. Partial fraction decomposition:
To integrate this expression, we need to decompose it into partial fractions.

Let's assume the decomposition as:
[1/2 (cos(9x) + u)]/(1 - 2u^3) = A/(1 - 2u) + B/(1 + u + u^2)

8. Find the values of A and B:
Multiply both sides by the denominator (1 - 2u)(1 + u + u^2) to get rid of the fractions.

1/2 (cos(9x) + u) = A(1 + u + u^2) + B(1 - 2u)

Comparing the coefficients of u^2, u, and constant terms, we can solve for A and B.

9. Perform the partial fraction decomposition:
Once we have the values of A and B, we can rewrite the expression using the partial fraction decomposition.

10. Integrating the partial fractions:
Now, we can integrate the partial fractions separately.

11. Substitute back the value of u:
After integrating, substitute back u = cos(x) to obtain the final result.

12. Write the final result:
The final result will be in terms of x after substituting back the value of u.

By following these steps, you can integrate the given expression.