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

2:44 minutes

Problem 8.RE.29

Textbook Question

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
29. ∫ cos⁴ x/sin⁶ x dx

Verified step by step guidance

Step 1: Recognize that the integral involves powers of trigonometric functions. Rewrite the integrand using trigonometric identities to simplify the expression. For example, use the identity

sin²x + cos²x = 1

if applicable.

Step 2: Break the integrand into separate terms or rewrite it in terms of a single trigonometric function. For instance, express

cos⁴x

(cos²x)²

and

sin⁶x

(sin²x)³

Step 3: Use substitution to simplify the integral. Let

u = sinx

, which implies

du = cosx dx

. Rewrite the integral in terms of

u

Step 4: After substitution, simplify the resulting integral. You may need to expand or factor terms to make the integration process easier. For example, the integral might involve powers of

u

that can be integrated directly.

Step 5: Once the integral is evaluated in terms of

u

, substitute back

u = sinx

to express the solution in terms of

x

. Simplify the final expression if necessary.

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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 simplifying complex integrals, such as those involving powers of trigonometric functions.

Trigonometric Identities

Trigonometric identities are equations that relate the angles and sides of triangles through sine, cosine, and other trigonometric functions. These identities, such as the Pythagorean identity and double angle formulas, can be used to simplify integrals involving trigonometric functions, making them easier to evaluate.

Power Reduction Formulas

Power reduction formulas are specific trigonometric identities that express powers of sine and cosine in terms of first-degree functions. For example, cos²(x) can be rewritten using the identity cos²(x) = (1 + cos(2x))/2. These formulas are particularly useful for integrals involving even powers of sine and cosine, as they help reduce the complexity of the integral.