9–61. Trigonometric integrals Evaluate the following integrals.
38. ∫ tan⁵θ sec⁴θ dθ

Verified step by step guidance

1. Step 1: Recognize that the integral involves powers of tangent and secant. For integrals involving these functions, it is often helpful to use trigonometric identities to simplify the expression. Recall the identity: $tan²\theta \; =\; sec²\theta \; -\; 1$.
2. Step 2: Split the powers of secant and tangent to facilitate substitution. Rewrite the integral as: $\int \; tan⁵\theta \; sec⁴\theta \; d\theta \; =\; \int \; tan⁵\theta \; sec²\theta \; sec²\theta \; d\theta$. This allows us to use substitution later.
3. Step 3: Use substitution. Let $u\; =\; tan\theta$, which implies $du\; =\; sec²\theta \; d\theta$. Replace $tan\theta$ and $sec²\theta \; d\theta$ in the integral.
4. Step 4: Rewrite the integral in terms of $u$. Using the substitution, the integral becomes: $\int \; u⁵\; sec²\theta \; d\theta \; =\; \int \; u⁵\; du$. This simplifies the problem to a basic polynomial integral.
5. Step 5: Integrate the polynomial. Apply the power rule for integration: $\int \; u⁵\; du\; =\; (u⁶)/6\; +\; C$. Finally, substitute back $u\; =\; tan\theta$ to express the result in terms of $\theta$.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, tangent, secant, and their inverses, are fundamental in calculus. They describe relationships between angles and sides of triangles and are periodic functions. Understanding their properties, such as identities and derivatives, is crucial for evaluating integrals involving these functions.

Integration Techniques

Integration techniques, including substitution and integration by parts, are essential for solving complex integrals. In the case of the integral ∫ tan⁵θ sec⁴θ dθ, recognizing patterns and using appropriate techniques can simplify the process. Mastery of these methods allows for the effective evaluation of integrals that may not be straightforward.

Secant and Tangent Identities

The secant and tangent functions are related through the identity sec²θ = 1 + tan²θ. This relationship is useful when integrating products of these functions, as it allows for substitutions that can simplify the integral. Understanding these identities is key to manipulating and solving integrals involving secant and tangent.