Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

7. Antiderivatives & Indefinite Integrals

Integrals of Trig Functions

Calculus

7. Antiderivatives & Indefinite Integrals

Integrals of Trig Functions

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3:05 minutes

Problem 8.3.28

Textbook Question

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

Verified step by step guidance

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.

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.

Step 3: Factor out the constant 6 from the integral to simplify the computation: ∫ 6 sec⁴x dx = 6 ∫ sec⁴x dx.

Step 4: Substitute the formula for ∫ sec⁴x dx into the expression: 6 ∫ sec⁴x dx = 6 [(1/3) tan³x + tan x + C].

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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.