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Calculus II Notes - Calculus II-Stewart Dr Berg Spring...

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Calculus II- Stewart Dr. Berg Spring 2010 Page 1 8.2 8.2 Trigonometric Integrals We use trigonometric identities to integrate certain combinations of trig functions. Powers of Sine and Cosine Example A Evaluate sin 3 x dx . Solution : We use the basic Pythagorean identity 1 = sin 2 x + cos 2 x to convert all but one copy of sine to cosine. Then sin 3 x dx = 1 cos 2 x ( ) sin x dx = sin x dx cos 2 x sin x dx = cos x + 1 3 cos 3 x + C . Example B Evaluate cos 2 x dx . Solution : Now we use the half-angle formula cos 2 x = 1 2 1 + cos2 x ( ) . Then cos 2 x dx = 1 2 1 + cos2 x ( ) dx = 1 2 dx + 1 2 cos2 x dx = x 2 + sin2 x 4 + C . These two examples illustrate the basic strategy for integrating products of powers of sine and cosine. Strategy for Products of Sine and Cosine a) If the power of the cosine is odd, save one cosine factor and use cos 2 x = 1 sin 2 x to convert the remaining factors of cosine to sine. Then substitute u = sin x . sin m x cos 2 k + 1 x dx = sin m x 1 sin 2 x ( ) k cos x dx b) If the power of the sine is odd, save one sine factor and use sin 2 x = 1 cos 2 x to convert the remaining factors of sine to cosine. Then substitute u = cos x .
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