Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Fundamentals of Trigonometric Identities
Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.
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### Calculus Exercise Problems #### Problem 15 Evaluate the integral: \[ \int \sin^3{x} \cos^4{x} \, dx \] #### Problem 16 Evaluate the integral: \[ \int \sec^4{x} \tan^4{x} \, dx \] ### Explanation: You are asked to solve the above integration problems. For Problem 15, use trigonometric identities and possibly a substitution method to simplify the integral involving sine and cosine. Similarly, for Problem 16, utilize the properties of secant and tangent trigonometric functions for integration. ### Tips for Solving: - For integrals involving powers of sine and cosine, a common technique is to use identities such as \( \sin^2{x} = 1 - \cos^2{x} \) to convert all terms to one trigonometric function type. - For integrals involving secant and tangent, look for opportunities to use substitutions involving \( u = \tan{x} \) or \( u = \sec{x} \).
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Transcribed Image Text:### Calculus Exercise Problems #### Problem 15 Evaluate the integral: \[ \int \sin^3{x} \cos^4{x} \, dx \] #### Problem 16 Evaluate the integral: \[ \int \sec^4{x} \tan^4{x} \, dx \] ### Explanation: You are asked to solve the above integration problems. For Problem 15, use trigonometric identities and possibly a substitution method to simplify the integral involving sine and cosine. Similarly, for Problem 16, utilize the properties of secant and tangent trigonometric functions for integration. ### Tips for Solving: - For integrals involving powers of sine and cosine, a common technique is to use identities such as \( \sin^2{x} = 1 - \cos^2{x} \) to convert all terms to one trigonometric function type. - For integrals involving secant and tangent, look for opportunities to use substitutions involving \( u = \tan{x} \) or \( u = \sec{x} \).
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