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

3:06 minutes

Problem 8.3.25

Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.
25. ∫ sin²x cos⁴x dx

Verified step by step guidance

Step 1: Recognize that the integral involves powers of sine and cosine. For integrals of this type, it is often helpful to use trigonometric identities to simplify the expression. Specifically, use the Pythagorean identity:

sin²x = 1 - cos²x

cos²x = 1 - sin²x

, depending on the situation.

Step 2: Split the powers of sine and cosine into manageable parts. For example, rewrite

sin²x cos⁴x

sin²x (cos²x)²

. This will allow us to work with smaller powers and apply substitution techniques.

Step 3: Use substitution to simplify the integral. Let

u = cosx

, which implies

du = -sinx dx

. Rewrite the integral in terms of

u

and

du

. The integral becomes

∫ sin²x cos⁴x dx = ∫ (1 - u²) u⁴ (-du)

Step 4: Simplify the integral further. Distribute the terms and simplify:

∫ (1 - u²) u⁴ (-du) = -∫ (u⁴ - u⁶) du

. Now, integrate each term separately using the power rule for integration:

∫ uⁿ du = uⁿ⁺¹ / (n + 1)

Step 5: After integrating, substitute back

u = cosx

to return the solution to the original variable

x

. Combine the terms and include 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. Key identities include the Pythagorean identities, angle sum and difference identities, and double angle formulas. These identities can simplify integrals involving trigonometric functions, making it easier to evaluate them.

Integration Techniques

Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and trigonometric substitution. For integrals involving products of trigonometric functions, such as sin²x and cos⁴x, using substitution or recognizing patterns can significantly simplify the process.

Power Reduction Formulas

Power reduction formulas are used to express higher powers of sine and cosine in terms of first powers. For example, sin²x can be rewritten using the identity sin²x = (1 - cos(2x))/2. These formulas are particularly useful in integrals, as they transform the integrand into a more manageable form, facilitating easier integration.