8.4 - 8.4 1 8.4 2 8.4 Trigonometric Integrals Products of...

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8.4 1 8.4 Trigonometric Integrals Products of Powers of Sines and Cosines We wish to evaluate integrals of the form: i sin m x cos n xdx where m and n are nonnegative integers. Recall the double angle formulas for the sine and cosine functions. sin 2 x = 2 sin x cos x cos 2 x = cos 2 x - sin 2 x The second formula can be used to to derive the very important “trig reduction” formulas. cos 2 x = 1 2 (1 + cos 2 x ) (1) sin 2 x = 1 2 (1 - cos 2 x ) (2) 8.4 2 These formulas are very useful when integrating even powers of sine and cosine. Example 1. Evaluate the following integrals. a. i cos 2 xdx By (1) we have i cos 2 xdx = 1 2 i (1 + cos 2 x ) dx = 1 2 p x + sin 2 x 2 P + C b. i sin 4 xdx Using reduction (twice) we have sin 4 x = ( sin 2 x ) 2 = 1 4 (1 - cos 2 x ) 2 = 1 4 ( 1 - 2 cos 2 x + cos 2 2 x ) = 1 4 p 1 - 2 cos 2 x + 1 2 (1 + cos 4 x ) P = 1 8 (3 - 4 cos 2 x + cos 4 x )
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8.4 3 It follows that i sin 4 xdx = 1 8 i (3 - 4 cos 2 x + cos 4 x ) dx = 1 8 p 3 x - 2 sin 2 x + sin 4 x 4 P + C Example 2. Even Products Evaluate i sin 4 x cos 6 xdx This is not too difficult since sin 4 x cos 6 x = ( sin 2 x ) 2
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This note was uploaded on 03/19/2008 for the course MATH 133 taught by Professor Wei during the Spring '07 term at Michigan State University.

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8.4 - 8.4 1 8.4 2 8.4 Trigonometric Integrals Products of...

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