2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
6. ∫ (2 − sin 2θ)/cos² 2θ dθ

Verified step by step guidance

1. Rewrite the integral by separating the terms in the numerator: ∫ (2 - sin(2θ)) / cos²(2θ) dθ = ∫ (2 / cos²(2θ)) dθ - ∫ (sin(2θ) / cos²(2θ)) dθ.
2. For the first term, recognize that 2 / cos²(2θ) can be rewritten using the secant function: 2 / cos²(2θ) = 2 sec²(2θ). The integral of sec²(x) is a standard result, so ∫ 2 sec²(2θ) dθ = (1/2) tan(2θ) + C.
3. For the second term, rewrite sin(2θ) / cos²(2θ) as (sin(2θ) / cos(2θ)) * (1 / cos(2θ)). Notice that (sin(2θ) / cos(2θ)) is equivalent to tan(2θ), and 1 / cos(2θ) is sec(2θ). Thus, the term becomes ∫ tan(2θ) sec(2θ) dθ.
4. To integrate ∫ tan(2θ) sec(2θ) dθ, use the substitution u = cos(2θ), which implies du = -2 sin(2θ) dθ. Rewrite the integral in terms of u and solve.
5. Combine the results of the two integrals, simplify, and include the constant of integration, C, to express the final solution.

Key Concepts

Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and partial fraction decomposition. Understanding these methods is crucial for evaluating complex integrals, as they allow for simplification and manipulation of the integrand to make integration feasible.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. Key identities, such as the Pythagorean identities and angle sum formulas, can simplify integrals involving trigonometric functions. Recognizing and applying these identities is essential for transforming the integrand into a more manageable form.

Substitution Method

The substitution method is a technique used in integration where a new variable is introduced to simplify the integral. By substituting a part of the integrand with a single variable, the integral can often be transformed into a standard form that is easier to evaluate. This method is particularly useful when dealing with composite functions or when the integrand contains a function and its derivative.