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)]
70. ∫ cos(x)cos(2x) dx

Verified step by step guidance

1. Step 1: Recognize that the integral ∫ cos(x)cos(2x) dx involves the product of two cosine functions. To simplify this, use the product-to-sum identity for cos(mx)cos(nx): cos(mx)cos(nx) = ½[cos((m-n)x) + cos((m+n)x)].
2. Step 2: Substitute m = 1 and n = 2 into the product-to-sum identity. This gives cos(x)cos(2x) = ½[cos((1-2)x) + cos((1+2)x)] = ½[cos(-x) + cos(3x)].
3. Step 3: Simplify further using the fact that cos(-x) = cos(x) (since cosine is an even function). This reduces the expression to cos(x)cos(2x) = ½[cos(x) + cos(3x)].
4. Step 4: Rewrite the integral using the simplified expression: ∫ cos(x)cos(2x) dx = ∫ ½[cos(x) + cos(3x)] dx. Factor out the constant ½: ∫ cos(x)cos(2x) dx = ½ ∫ [cos(x) + cos(3x)] dx.
5. Step 5: Split the integral into two separate integrals: ∫ cos(x)cos(2x) dx = ½ [∫ cos(x) dx + ∫ cos(3x) dx]. Now, integrate each term separately. For ∫ cos(x) dx, the result is sin(x). For ∫ cos(3x) dx, use the substitution u = 3x, du = 3 dx, which leads to (1/3)sin(3x). Combine these results to complete the solution.

Key Concepts

Product-to-Sum Identities

Product-to-sum identities are trigonometric formulas that express the product of sine and cosine functions as a sum of sine or cosine functions. These identities simplify the integration of products of trigonometric functions by transforming them into a more manageable form. For example, the identity for sin(mx)cos(nx) allows us to rewrite the integral in a way that is easier to evaluate.

Integration of Trigonometric Functions

Integrating trigonometric functions involves finding the antiderivative of functions like sine and cosine. The basic integrals, such as ∫sin(x)dx = -cos(x) + C and ∫cos(x)dx = sin(x) + C, are foundational. Understanding these integrals is crucial when applying product-to-sum identities, as the transformed functions will often be simpler trigonometric functions that can be integrated directly.

Definite and Indefinite Integrals

Definite integrals calculate the area under a curve between two limits, while indefinite integrals represent a family of functions without specific bounds. In the context of the given question, recognizing whether the integral is definite or indefinite is important for applying the correct evaluation techniques and understanding the final result. The integral ∫cos(x)cos(2x)dx, for instance, is an indefinite integral that will yield a general antiderivative.