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

3:07 minutes

Problem 8.3.26

Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.
26. ∫ sin³x cos³ᐟ²x dx

Verified step by step guidance

Rewrite the integral using trigonometric identities. Specifically, use the identity for powers of sine and cosine: \( \sin^3(x) = \sin(x) \cdot \sin^2(x) \) and \( \sin^2(x) = 1 - \cos^2(x) \). This transforms the integral into \( \int \sin(x) (1 - \cos^2(x)) \cos^{3/2}(x) \, dx \).

Perform a substitution to simplify the integral. Let \( u = \cos(x) \), which implies \( du = -\sin(x) \, dx \). Substitute these into the integral, replacing \( \sin(x) \, dx \) with \( -du \) and \( \cos(x) \) with \( u \). The integral becomes \( -\int (1 - u^2) u^{3/2} \, du \).

Expand the integrand \( (1 - u^2) u^{3/2} \) by distributing \( u^{3/2} \). This gives \( -\int (u^{3/2} - u^{7/2}) \, du \).

Integrate each term separately. Use the power rule for integration, \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \), to find the antiderivative of \( u^{3/2} \) and \( u^{7/2} \).

After finding the antiderivative, substitute back \( u = \cos(x) \) to return to the original variable. Simplify the expression to complete the solution.

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.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. They are essential for simplifying integrals involving trigonometric functions. Common identities include the Pythagorean identities, angle sum and difference identities, and double angle formulas. Understanding these identities helps in transforming complex integrals into simpler forms that are easier to evaluate.

Integration Techniques

Integration techniques are methods used to find the integral of a function. For trigonometric integrals, techniques such as substitution, integration by parts, and the use of trigonometric identities are often employed. In the case of products of sine and cosine functions, using identities to express the integrand in a more manageable form can significantly simplify the integration process.

Power Reduction Formulas

Power reduction formulas are specific trigonometric identities that allow us to express powers of sine and cosine in terms of first-degree functions. For example, sin²x can be rewritten using the identity sin²x = (1 - cos(2x))/2. These formulas are particularly useful when integrating higher powers of sine and cosine, as they reduce the degree of the functions, making the integral easier to solve.

Watch next

Master Integrals Resulting in Basic Trig Functions with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

Textbook Question

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

8. ∫ sin 3x cos 2x dx

views