2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
22. ∫ tan³ 5θ dθ

Verified step by step guidance

1. Rewrite the integral by separating the tangent function into a product of tan²(5θ) and tan(5θ): ∫ tan³(5θ) dθ = ∫ tan²(5θ) tan(5θ) dθ.
2. Use the trigonometric identity tan²(x) = sec²(x) - 1 to rewrite tan²(5θ): ∫ tan²(5θ) tan(5θ) dθ = ∫ (sec²(5θ) - 1) tan(5θ) dθ.
3. Let u = sec(5θ), so that du = 5 sec(5θ) tan(5θ) dθ. Solve for tan(5θ) dθ in terms of du: tan(5θ) dθ = (1/5) du.
4. Substitute u = sec(5θ) and tan(5θ) dθ = (1/5) du into the integral: ∫ (sec²(5θ) - 1) tan(5θ) dθ = (1/5) ∫ (u² - 1) du.
5. Integrate (u² - 1) with respect to u: (1/5) ∫ (u² - 1) du = (1/5) [(u³/3) - u] + C. Finally, substitute back u = sec(5θ) to express the result in terms of θ.

Key Concepts

Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and trigonometric identities. Understanding these methods is crucial for solving complex integrals, as they allow for the simplification of the integrand into a more manageable form.

Trigonometric Identities

Trigonometric identities are equations that relate the angles and sides of triangles through sine, cosine, tangent, and their reciprocals. These identities, such as the Pythagorean identity and angle sum formulas, are essential for simplifying integrals involving trigonometric functions, like tan³(5θ), making them easier to integrate.

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 complicated expressions.