**Integration of x cos(x) cos(2x)**

To find the integral of the expression x cos(x) cos(2x), we can use the method of integration by parts. Integration by parts is a technique based on the product rule of differentiation, which allows us to integrate certain products of functions.

The formula for integration by parts is given as follows:

∫ u dv = uv - ∫ v du

where u and v are functions of x, and du and dv are their respective derivatives.

Now, let's proceed with the integration of x cos(x) cos(2x) using integration by parts.

**Step 1:** Identify the functions u and dv.

In this case, we can choose u = x and dv = cos(x) cos(2x).

**Step 2:** Calculate the derivatives du and v.

Taking the derivative of u = x yields du = dx.

To find v, we need to integrate dv = cos(x) cos(2x). This requires the use of trigonometric identities. Using the double-angle formula for cosine, we can rewrite cos(2x) as 2cos^2(x) - 1.

Therefore, dv = cos(x) (2cos^2(x) - 1) dx.

Integrating dv, we get v = ∫ cos(x) (2cos^2(x) - 1) dx.

**Step 3:** Apply the integration by parts formula.

Using the formula ∫ u dv = uv - ∫ v du, we can rewrite the integral as:

∫ x cos(x) cos(2x) dx = x v - ∫ v du

Substituting the values we obtained for u, v, du, and dv, we have:

∫ x cos(x) cos(2x) dx = x (∫ cos(x) (2cos^2(x) - 1) dx) - ∫ (∫ cos(x) (2cos^2(x) - 1) dx) dx

**Step 4:** Simplify and solve the integrals.

Now, we need to evaluate the integrals. The first integral on the right-hand side can be solved using integration by parts again, and the second integral can be evaluated using trigonometric identities and basic integration formulas.

By following these steps, you will be able to integrate the expression x cos(x) cos(2x) and find the final result.

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