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

Previous problemNext problem

2:11 minutes

Problem 8.RE.48

Textbook Question

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
48. ∫ sin(3x) cos⁶(3x) dx

Verified step by step guidance

Step 1: Recognize that the integral involves a product of trigonometric functions, specifically sin(3x) and cos⁶(3x). To simplify, consider using a substitution method. Let u = cos(3x), which implies that du = -3sin(3x) dx.

Step 2: Rewrite the integral in terms of u. Substitute sin(3x) dx with -du/3 and cos⁶(3x) with u⁶. The integral becomes: ∫ sin(3x) cos⁶(3x) dx = -1/3 ∫ u⁶ du.

Step 3: Apply the power rule for integration to evaluate ∫ u⁶ du. Recall that the power rule states: ∫ uⁿ du = uⁿ⁺¹ / (n+1) + C, where n ≠ -1.

Step 4: Substitute back u = cos(3x) into the result obtained from the integration step to express the solution in terms of x.

Step 5: Add the constant of integration (C) to complete the solution, as this is an indefinite integral.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

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 essential for evaluating complex integrals, as they allow for simplification and manipulation of the integrand to facilitate integration.

Trigonometric Identities

Trigonometric identities are equations that relate the angles and sides of triangles, and they are crucial in simplifying integrals involving trigonometric functions. For example, the product-to-sum identities can transform products of sine and cosine into sums, making integration more manageable. Familiarity with these identities is vital for solving integrals like ∫ sin(3x) cos⁶(3x) dx.

Substitution Method

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