9–61. Trigonometric integrals Evaluate the following integrals.
28. ∫ 6 sec⁴x dx

Verified step by step guidance

1. Step 1: Recognize the integral ∫ 6 sec⁴x dx as a trigonometric integral involving secant raised to the fourth power. The goal is to simplify and evaluate this integral.
2. Step 2: Recall the standard formula for the integral of sec⁴x: ∫ sec⁴x dx = (1/3) tan³x + tan x + C. This formula can be applied directly to the given integral.
3. Step 3: Factor out the constant 6 from the integral to simplify the computation: ∫ 6 sec⁴x dx = 6 ∫ sec⁴x dx.
4. Step 4: Substitute the formula for ∫ sec⁴x dx into the expression: 6 ∫ sec⁴x dx = 6 [(1/3) tan³x + tan x + C].
5. Step 5: Simplify the expression by distributing the constant 6 across the terms inside the brackets: 6 [(1/3) tan³x + tan x + C] = 2 tan³x + 6 tan x + C.

Key Concepts

Integration of Trigonometric Functions

Integration of trigonometric functions involves finding the antiderivative of functions that include trigonometric ratios. In this case, the integral of secant raised to a power, such as sec⁴x, requires knowledge of specific integration techniques and formulas related to trigonometric identities.

Secant Function

The secant function, denoted as sec(x), is the reciprocal of the cosine function, expressed as sec(x) = 1/cos(x). Understanding the properties and behavior of the secant function is crucial for evaluating integrals involving secant, especially when raised to higher powers, as it can often be simplified using trigonometric identities.

Integration Techniques

Various techniques exist for integrating functions, including substitution, integration by parts, and recognizing patterns in integrals. For sec⁴x, one might use the identity sec²x = 1 + tan²x to rewrite the integral, making it easier to evaluate. Familiarity with these techniques is essential for solving complex integrals.