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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3:24 minutes

Problem 8.3.68

Textbook Question

67-70. Integrals of the form ∫ sin(mx)cos(nx) dx Use the following product-to-sum identities to evaluate the given integrals:
sin(mx)sin(nx) = ½[cos((m-n)x) - cos((m+n)x)]
sin(mx)cos(nx) = ½[sin((m-n)x) + sin((m+n)x)]
cos(mx)cos(nx) = ½[cos((m-n)x) + cos((m+n)x)]
68. ∫ sin(5x)sin(7x) dx

Verified step by step guidance

Step 1: Recognize that the integral involves the product of two sine functions, sin(5x) and sin(7x). To simplify this, use the product-to-sum identity for sin(mx)sin(nx): sin(mx)sin(nx) = ½[cos((m-n)x) - cos((m+n)x)].

Step 2: Substitute m = 5 and n = 7 into the identity. This gives sin(5x)sin(7x) = ½[cos((5-7)x) - cos((5+7)x)]. Simplify the expressions inside the cosine functions: (5-7)x = -2x and (5+7)x = 12x.

Step 3: Rewrite the integral using the product-to-sum identity: ∫ sin(5x)sin(7x) dx = ½ ∫ [cos(-2x) - cos(12x)] dx. Note that cos(-2x) = cos(2x) because cosine is an even function.

Step 4: Break the integral into two separate integrals: ∫ sin(5x)sin(7x) dx = ½ [∫ cos(2x) dx - ∫ cos(12x) dx].

Step 5: Integrate each term separately. Recall that the integral of cos(kx) is (1/k)sin(kx) + C, where k is a constant. Apply this formula to both ∫ cos(2x) dx and ∫ cos(12x) dx, and combine the results to complete the solution.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Product-to-Sum Identities

Product-to-sum identities are trigonometric formulas that express products of sine and cosine functions as sums or differences of sine and cosine functions. These identities simplify the integration of products of trigonometric functions by transforming them into a more manageable form, making it easier to evaluate integrals.

Integration Techniques

Integration techniques refer to various methods used to compute integrals, including substitution, integration by parts, and the use of trigonometric identities. In the context of the given question, applying product-to-sum identities is a specific technique that allows for the simplification of integrals involving products of sine and cosine functions.

Definite vs. Indefinite Integrals

Definite integrals calculate the area under a curve between two specified limits, while indefinite integrals represent a family of functions and include a constant of integration. Understanding the difference is crucial when evaluating integrals, as it affects the final result and the interpretation of the integral's meaning in a given context.