91Ó°ÊÓ

Understanding trigonometric identities can greatly simplify integration, especially when dealing with products of trigonometric functions. These identities are like shortcuts in your mathematical toolkit. They allow you to transform complicated expressions into simpler forms that are often easier to integrate. An important identity used in this exercise is the product-to-sum identity for sine functions:\[ \sin A \sin B = \frac{1}{2} [\cos(A-B) - \cos(A+B)] \]This identity converts the product of two sine functions into a difference of cosine functions, making the integral more manageable. For our specific problem, substituting \(A = 4t\) and \(B = \frac{t}{2}\) simplifies the original expression dramatically, enabling the integration process to begin. Recognizing and applying these identities is a critical skill when working with trigonometric integrals.

The choice of integration techniques defines how efficiently and accurately you can solve an integral. In this case, we focused on using a transformation technique involving trigonometric identities. Once we applied the product-to-sum formula, the integrand was expressed as a combination of cosine functions:\[ 8 \int \sin(4t) \sin\left(\frac{t}{2}\right) \, dt = 4 \int \left[ \cos\left(\frac{7t}{2}\right) - \cos\left(\frac{9t}{2}\right) \right] \, dt\]Next, the integral of each cosine term was computed separately. This separation is a common integration technique where you handle each part of the function individually:- For \( \int \cos\left(\frac{7t}{2}\right) \, dt \), we find \( \frac{2}{7} \sin\left(\frac{7t}{2}\right) + C_1 \).- For \( \int \cos\left(\frac{9t}{2}\right) \, dt \), we find \( \frac{2}{9} \sin\left(\frac{9t}{2}\right) + C_2 \).Integrating cosine functions follows a direct process: use the antiderivative formula \( \int \cos(x) \, dx = \sin(x) + C \). Substituting these results back into the simplified expression allows us to combine terms to find the final integral.

Product-to-sum formulas are particularly useful in transforming integrals involving products of trigonometric functions into forms that are straightforward to work with. These transformations are invaluable in integration because they enable the separation of terms, as seen in this problem.Typically, these formulas convert products of sines or cosines into sums or differences of simpler trigonometric functions. The formula used here for sine is a classic example:\[ \sin A \sin B = \frac{1}{2} [\cos(A-B) - \cos(A+B)] \]By applying it, we changed a product of two sines into a palindrome-like expression involving cosine. This method not only simplifies the process but also reduces potential errors when solving integrals. Our original integral \( 8 \sin(4t) \sin\left(\frac{t}{2}\right) \) became much easier to handle once the product-to-sum identity was applied. The challenge of converting difficult products into simple sums is what makes these formulas a powerful technique in solving integrals involving trigonometric functions.