67-70. Integrals of the form ∫ sin(mx)cos(nx) dx Use the following product-to-sum identities to evaluate the given integrals:
sin(mx)sin(nx) = ½[cos((m-n)x) - cos((m+n)x)]
sin(mx)cos(nx) = ½[sin((m-n)x) + sin((m+n)x)]
cos(mx)cos(nx) = ½[cos((m-n)x) + cos((m+n)x)]
68. ∫ sin(5x)sin(7x) dx

Verified step by step guidance

1. Step 1: Recognize that the integral involves the product of two sine functions, sin(5x) and sin(7x). To simplify this, use the product-to-sum identity for sin(mx)sin(nx): sin(mx)sin(nx) = ½[cos((m-n)x) - cos((m+n)x)].
2. Step 2: Substitute m = 5 and n = 7 into the identity. This gives sin(5x)sin(7x) = ½[cos((5-7)x) - cos((5+7)x)]. Simplify the expressions inside the cosine functions: (5-7)x = -2x and (5+7)x = 12x.
3. Step 3: Rewrite the integral using the product-to-sum identity: ∫ sin(5x)sin(7x) dx = ½ ∫ [cos(-2x) - cos(12x)] dx. Note that cos(-2x) = cos(2x) because cosine is an even function.
4. Step 4: Break the integral into two separate integrals: ∫ sin(5x)sin(7x) dx = ½ [∫ cos(2x) dx - ∫ cos(12x) dx].
5. Step 5: Integrate each term separately. Recall that the integral of cos(kx) is (1/k)sin(kx) + C, where k is a constant. Apply this formula to both ∫ cos(2x) dx and ∫ cos(12x) dx, and combine the results to complete the solution.

Key Concepts

Product-to-Sum Identities

Product-to-sum identities are trigonometric formulas that express products of sine and cosine functions as sums or differences of sine and cosine functions. These identities simplify the integration of products of trigonometric functions by transforming them into a more manageable form, making it easier to evaluate integrals.

Integration Techniques

Integration techniques refer to various methods used to compute integrals, including substitution, integration by parts, and the use of trigonometric identities. In the context of the given question, applying product-to-sum identities is a specific technique that allows for the simplification of integrals involving products of sine and cosine functions.

Definite vs. Indefinite Integrals

Definite integrals calculate the area under a curve between two specified limits, while indefinite integrals represent a family of functions and include a constant of integration. Understanding the difference is crucial when evaluating integrals, as it affects the final result and the interpretation of the integral's meaning in a given context.