8.4.7-eg-1 - Example Evaluate cos6 x dx. 2 When the power...

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Unformatted text preview: Example Evaluate cos6 x dx. 2 When the power of sine or cosine is even, we write the integrand as the power of a square and use a double angle formula: cos6 x x = cos2 2 2 3 = 1 + cos 2 · x 2 2 3 = 1 (1 + cos(x))3 . 8 Expand the power and apply the double angle formula for the even power again: (1 + cos x)3 = 1 + 3 cos x + 3 cos2 x + cos3 x 1 + cos(2x) = 1 + 3 cos x + 3 + cos3 x. 2 Now integrate term by term, making the substitution u = sin x in the last integral, cos6 x 2 dx = 1 8 1 + 3 cos x + 3 1 + cos(2x) + cos3 x dx 2 = 1 3 3 x + 3 sin x + x + sin(2x) + 8 2 4 = 15 3 x + 3 sin x + sin(2x) + 82 4 = 15 3 1 x + 3 sin x + sin(2x) + sin x − sin3 x + C 82 4 3 = 5 1 3 1 x + sin x + sin(2x) − sin3 x + C. 16 2 32 24 (1 − sin2 x) cos x dx (1 − u2 ) du ...
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This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.

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