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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2:59 minutes

Problem 8.3.15

Textbook Question

9–61. Trigonometric integrals Evaluate the following integrals.
15. ∫ sin³x cos²x dx

Verified step by step guidance

Step 1: Recognize that the integral involves powers of sine and cosine. To simplify, use trigonometric identities. Specifically, split the odd power of sine: sin³x = sin²x * sinx, and recall the Pythagorean identity sin²x = 1 - cos²x.

Step 2: Rewrite the integral as ∫ (1 - cos²x) * sinx * cos²x dx. This allows us to express the integral in terms of cosx, making substitution easier.

Step 3: Perform a substitution. Let u = cosx, which implies that du = -sinx dx. Substitute these into the integral, replacing cosx with u and sinx dx with -du.

Step 4: After substitution, the integral becomes ∫ (1 - u²) * u² * (-du). Simplify this expression to ∫ (-u² + u⁴) du.

Step 5: Integrate term by term. Use the power rule for integration: ∫ uⁿ du = uⁿ⁺¹ / (n+1). After integrating, back-substitute u = cosx to express the result in terms of x.

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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. They are essential for simplifying integrals involving trigonometric functions. For example, the Pythagorean identity sin²x + cos²x = 1 can be used to rewrite integrals in a more manageable form.

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 the integral ∫ sin³x cos²x dx, using substitution or recognizing patterns in the integrand can simplify the process of finding the antiderivative.

Power Reduction Formulas

Power reduction formulas are used to express powers of sine and cosine in terms of first-degree functions. These formulas, such as sin²x = (1 - cos(2x))/2, help in simplifying integrals involving higher powers of trigonometric functions. Applying these formulas can make the integration of functions like sin³x cos²x more straightforward.

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