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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3:03 minutes

Problem 8.3.11

Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.
11. ∫ sin²(3x) dx

Verified step by step guidance

Step 1: Recognize that the integral involves a squared trigonometric function, sin²(3x). To simplify, use the trigonometric identity: sin²(u) = (1 - cos(2u)) / 2. Substitute u = 3x into the identity.

Step 2: Rewrite the integral using the identity: ∫ sin²(3x) dx = ∫ (1 - cos(6x)) / 2 dx. This simplifies the integral into two separate terms.

Step 3: Split the integral into two parts: ∫ sin²(3x) dx = (1/2) ∫ 1 dx - (1/2) ∫ cos(6x) dx. Now, handle each term individually.

Step 4: Evaluate the first term, (1/2) ∫ 1 dx, which is straightforward and results in (1/2)x. For the second term, (1/2) ∫ cos(6x) dx, use the substitution method. Let u = 6x, then du = 6 dx, and rewrite the integral accordingly.

Step 5: After substitution, integrate cos(u) to get sin(u). Substitute back u = 6x into the result. Combine the results from both terms to express the final solution, adding the constant of integration C.

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Key Concepts

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

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. A key identity for integrating sine squared functions is the Pythagorean identity, which states that sin²(x) = (1 - cos(2x))/2. This identity simplifies the integration process by transforming the sine function into a form that is easier to integrate.

Integration Techniques

Integration techniques are methods used to find the integral of a function. For the integral of sin²(3x), one common technique is substitution, where we can let u = 3x, making du = 3dx. This allows us to rewrite the integral in terms of u, simplifying the integration process and making it more manageable.

Definite vs. Indefinite Integrals

Definite and indefinite integrals are two types of integrals in calculus. An indefinite integral, like ∫ sin²(3x) dx, represents a family of functions and includes a constant of integration (C). In contrast, a definite integral calculates the area under the curve between two specific limits. Understanding the difference is crucial for correctly interpreting the results of integration.

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