91Ó°ÊÓ

Product-to-sum identities are extremely useful in simplifying certain integrals, particularly those involving products of trigonometric functions. These identities transform products of sine and cosine into sums or differences of sines, making integration much more approachable. Here’s the key identity:

• \(\sin A \cos B = \frac{1}{2} [\sin(A+B) + \sin(A-B)]\)

Let's go through this step-by-step in the context of the given problem.To transform \( \sin 3x \cos \frac{1}{2}x \), we assign:

• \( A = 3x \)
• \( B = \frac{1}{2}x \)

Using the product-to-sum identity, the expression becomes \( \frac{1}{2} [\sin(\frac{7}{2}x) + \sin(\frac{5}{2}x)] \).By expressing the product as a sum of sines, the function becomes easier to integrate, turning a complex product into simpler components.

Trigonometric integrals often involve functions of sine and cosine. These integrals can sometimes appear complex but can be handled with identities from trigonometry. For example, after applying the product-to-sum identities, we reduce our integral into parts that are standard and easier to manage.

• Original function: \( \int \sin 3x \cos \frac{1}{2}x \, dx \)
• Transformed function: \( \int \frac{1}{2} [\sin(\frac{7}{2}x) + \sin(\frac{5}{2}x)] \, dx \)

Now, each term within the integral is a simpler trigonometric integral. For sine integrals, remember:

• \( \int \sin kx \, dx = -\frac{1}{k} \cos kx + C \)

Recognizing this pattern helps evaluate these integrals with greater ease.

Trigonometric integrals like the one we're dealing with often need specific integration techniques to solve them effectively.

• The primary technique involves recognizing identities to simplify the integral structure.
• In our exercise, this was achieved by transforming the product using the product-to-sum identity.
• Next, break down the integral into smaller, standard parts that you can integrate directly.

For this problem, after applying the product-to-sum conversion:

• The integral becomes: \( \frac{1}{2} \left( \int \sin (\frac{7}{2}x) \ dx + \int \sin (\frac{5}{2}x) \ dx \right) \)
• Each term is decomposed using the antiderivative of sine, making the expressions straightforward to handle.

These techniques illustrate how a seemingly complicated integral can be broken down into manageable steps using strategic approaches.