7.2 Trigonometric Integrals

7.2 Trigonometric Integrals - 7.2 Trigonometric Integrals...

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7.2 Trigonometric Integrals In this section, we introduce techniques for evaluating integrals of the form ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 dx x x dx x x dx x x n m n m n m csc cot sec tan cos sin where either m or n is a nonnegative integer and occasionally, integrals involving the product of sines and cosines of two different angles. REFER to the Handout for Trigonometric Integrals
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Integrals Involving Sine and Cosine dx x x n m ) ( cos ) ( sin A. 1. If the power of the sine is odd and positive, save one sine factor and convert the remaining factors to cosine. Then, expand and integrate. ( 29 ( 29 xdx x x xdx x x xdx x n k n k n k sin cos cos 1 sin cos sin cos sin 2 2 1 2 - = = + odd convert to cosine save for du
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1. If the power of the cosine is odd and positive, save one cosine factor and convert the remaining factors to sine. Then, expand and integrate. ( 29 ( 29 xdx x x xdx x x xdx x k m k m k m sin sin 1 sin cos cos sin cos sin 2 2 1 2 - = = + odd convert to sine save for du 1. If the powers of both the sine and cosine are even and nonnegative, make repeated use of the identities to convert the integrand to odd powers of the cosine. Then, proceed as in case 2. It is sometimes helpful to use the identity 2 2 cos 1 sin 2 x x - = and 2 2 cos 1 cos 2 x x + = x x x 2 sin 2 1 cos sin =
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Examples: Integrals Involving Sine and Cosine Evaluate the integral.
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