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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2:16 minutes

Problem 8.RE.32

Textbook Question

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
32. ∫ csc²(6x) cot(6x) dx

Verified step by step guidance

Step 1: Recognize the integral ∫ csc²(6x) cot(6x) dx as a product of trigonometric functions. Notice that csc²(6x) is the derivative of cot(6x). This suggests a substitution method might be effective.

Step 2: Let u = cot(6x). Then, compute the derivative of u with respect to x: du/dx = -6 csc²(6x). Rearrange to express dx in terms of du: dx = -du / (6 csc²(6x)).

Step 3: Substitute u = cot(6x) and dx = -du / (6 csc²(6x)) into the integral. The csc²(6x) term in the original integral will cancel out, leaving ∫ u (-du / 6).

Step 4: Simplify the integral to ∫ (-u / 6) du. Factor out the constant -1/6 to get (-1/6) ∫ u du.

Step 5: Integrate u with respect to u. The integral of u is u²/2. Multiply by the constant -1/6 to get (-1/6)(u²/2). Substitute back u = cot(6x) to express the result in terms of x.

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Key Concepts

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

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 integrals that may not be straightforward, as they allow for simplification and manipulation of the integrand.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and their derivatives, play a significant role in calculus. In this problem, csc²(6x) and cot(6x) are trigonometric functions that can be integrated using their known derivatives. Recognizing the relationships between these functions is essential for applying the appropriate integration techniques.

Substitution Method

The substitution method is a powerful technique in integration that involves changing the variable of integration to simplify the integral. By substituting a part of the integrand with a new variable, the integral can often be transformed into a more manageable form. This method is particularly useful when dealing with composite functions or when the integrand contains a function and its derivative.