Integrate: `intcosm xcosn x\ dx ,\ \ m!=n`

Explanation

To solve the integral \( \int \cos(mx) \cos(nx) \, dx \) where \( m \neq n \), we can use the product-to-sum identities for cosine functions. Here’s a step-by-step solution:

### Step 1: Apply the Product-to-Sum Formula
The product-to-sum formula states that:
\[ \cos A \cos B = \frac{1}{2} \left( \cos(A + B) + \cos(A - B) \right) \]
In our case, let \( A = mx \) and \( B = nx \). Thus, we have:
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