First, we can use the product-to-sum formula for sine functions, which states that:
$$\sin{A} \sin{B} = \frac{1}{2}(\cos{(A-B)} - \cos{(A+B)})$$
Applying this formula to our integral, we get:
$$\int \sin{5x} \sin{4x} dx = \int \frac{1}{2}(\cos{(5x-4x)} -
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