9–61. Trigonometric integrals Evaluate the following integrals.
34. ∫ tan⁹x sec⁴x dx

Verified step by step guidance

1. Step 1: Recognize that the integral involves powers of tangent and secant. For trigonometric integrals involving these functions, it is often helpful to use trigonometric identities to simplify the expression. Recall the identity: $sec²x\; =\; 1\; +\; tan²x$.
2. Step 2: Split the powers of secant and tangent to facilitate substitution. Specifically, reserve one $sec²x$ term for substitution, as $d(tanx)\; =\; sec²x\; dx$. Rewrite the integral as: $\int \; tan⁹x\; sec²x\; sec²x\; dx$.
3. Step 3: Perform substitution. Let $u\; =\; tanx$, which implies $du\; =\; sec²x\; dx$. Replace $tanx$ with $u$ and $sec²x\; dx$ with $du$. The integral becomes: $\int \; u⁹\; sec²x\; du$.
4. Step 4: Simplify further using the substitution. Since $sec²x$ has already been accounted for in the substitution, the integral reduces to: $\int \; u⁹\; du$. This is a straightforward power rule integration.
5. Step 5: Apply the power rule for integration. Recall that the integral of $uⁿ$ is $(uⁿ⁺¹)/(n+1)\; +\; C$, where $C$ is the constant of integration. After integrating, substitute back $u\; =\; tanx$ to express the result in terms of $x$.

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 ∫ tan⁹x sec⁴x dx, recognizing patterns and using appropriate substitutions can simplify the process. Mastery of these techniques 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 sec²x = 1 + tan²x. This relationship is particularly useful in integrals involving powers of tangent and secant, as it allows for the conversion between the two functions. Utilizing these identities can simplify the integration process and lead to a more manageable expression.