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Integration using trigonometric identities
- Lesson: 1a11:15
- Lesson: 2a16:35
- Lesson: 3a9:49
Integration using trigonometric identities
In this section, we will take a look at several methods for integrating trigonometric functions. All methods require us to use U-Substitution and substituting with trigonometric identities. In addition, the trigonometric functions we are dealing with are products of sine and cosine with powers. There are a total of 3 cases. The first case is when the power of cosine is odd. The second case is when the power of sine is odd. Lastly, the third case is when both the powers of sine and cosine are even.
Basic Concepts: Other word problems relating angles in trigonometry, Double-angle identities, Pythagorean identities, Quotient identities and reciprocal identities, Sum and difference identities, Cofunction identities
Lessons
Pre-requisite: * Trigonometry –"Trigonometric Identities"
Note: Strategy for evaluating ∫sin■x cos■x dx
Case 1: ∫sin■x cosoddx dx
1. strip out one cosine factor
2. express the remaining cosine factors in terms of sine using the Pythagorean Identity: cos2x=1−sin2x
3. substitute u=sinx
Case 2: ∫sinoddx cos■x dx
1. strip out one sine factor
2. express the remaining sine factors in terms of cosine using the Pythagorean Identity: sin2x=1−cos2x
3. substitute u=cosx
Case 3: ∫sinevenx cosevenx dx
1. use the half-angle identities: sin2x=21(1−cos2x) or cos2x=21(1+cos2x)
2. if necessary, use the double-angle identity: sinx cosx =21sin2x
Note: Strategy for evaluating ∫sin■x cos■x dx
Case 1: ∫sin■x cosoddx dx
1. strip out one cosine factor
2. express the remaining cosine factors in terms of sine using the Pythagorean Identity: cos2x=1−sin2x
3. substitute u=sinx
Case 2: ∫sinoddx cos■x dx
1. strip out one sine factor
2. express the remaining sine factors in terms of cosine using the Pythagorean Identity: sin2x=1−cos2x
3. substitute u=cosx
Case 3: ∫sinevenx cosevenx dx
1. use the half-angle identities: sin2x=21(1−cos2x) or cos2x=21(1+cos2x)
2. if necessary, use the double-angle identity: sinx cosx =21sin2x
- 1.Evaluate the integral (odd power of cosine).a)∫sin8x cos3x dx
- 2.Evaluate the integral (odd power of sine).a)∫sin5x cos2x dx
- 3.Evaluate the integral (even powers of sine/cosine).a)∫sin2x dx