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Chapter 7 – Techniques of Integration

Chapter 7 – Techniques of Integration. 7.2 Trigonometric Integrals. Trigonometric Integrals. Consider evaluating the following integral: For this and the following cases we will need to use the trigonometric identities to integrate certain combinations of trigonometric functions. Examples.

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Chapter 7 – Techniques of Integration

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  1. Chapter 7 – Techniques of Integration 7.2 Trigonometric Integrals 7.2 Trigonometric Integrals

  2. Trigonometric Integrals • Consider evaluating the following integral: • For this and the following cases we will need to use the trigonometric identities to integrate certain combinations of trigonometric functions. 7.2 Trigonometric Integrals

  3. Examples • Evaluate the integral 7.2 Trigonometric Integrals

  4. Strategy for Evaluating Part a • If the power of cosine is odd (n = 2k + 1), save one cosine factor and use cos2x = 1 – sin2x to express the remaining factors in terms of sine: • Then substitute u = sinx. 7.2 Trigonometric Integrals

  5. Strategy for Evaluating Part b • If the power of sine is odd (n = 2k + 1), save one sine factor and use sin2x = 1 – cos2x to express the remaining factors in terms of sine: • Then substitute u = cosx. • NOTE: If both sine and cosine are odd then either part a or part b can be used. 7.2 Trigonometric Integrals

  6. Strategy for Evaluating Part c • If the powers of both sine and cosine are even, use the half-angle identities • It is sometimes helpful to use the identity 7.2 Trigonometric Integrals

  7. Example 2 • Evaluate the integral. 7.2 Trigonometric Integrals

  8. Strategy for Evaluating Part a • If the power of secant is even (n = 2k, k ≥ 2), save a factor of sec2x and use sec2x = 1 + tan2x to express the remaining factors in terms of tangent: • Then substitute u = tanx. 7.2 Trigonometric Integrals

  9. Strategy for Evaluating Part a • If the power of tangent is odd (n = 2k +1), save a factor of secxtanx and use tan2x = sec2x – 1 to express the remaining factors in terms of secant: • Then substitute u = sec x. 7.2 Trigonometric Integrals

  10. Example 4 • Evaluate the integral. 7.2 Trigonometric Integrals

  11. Other hints • Sometimes we may be able to integrate using the following formulas: 7.2 Trigonometric Integrals

  12. Example 5 • Evaluate the integral. 7.2 Trigonometric Integrals

  13. Other Trigonometric Identities To evaluate the integrals (a) , (b) , or (c) , use the corresponding identity: 7.2 Trigonometric Integrals

  14. Example 6 – Page 466 #53 • Ignore the book directions and just evaluate the integral. 7.2 Trigonometric Integrals

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