Trig_integrals_slides

Trig_integrals_slides - Section 8.3 Trigonometric Integrals...

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Section 8.3: Trigonometric Integrals
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cos a ( x )sin b ( x ) dx ! Done by substitution: u= sin(x) or u=cos(x). how do we pick the right u?
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cos( x )sin 6 ( x ) dx ! A) u= sin(x) B) u=cos(x)
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I = cos( x )sin 6 ( x ) dx ! A) u= sin(x) du= cos(x)dx I = u 6 du = ! u 7 7 + c = sin 7 ( x ) 7 + c
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cos 4 ( x )sin 5 ( x ) dx ! A) u= sin(x) B) u=cos(x)
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If the power of cosine is odd , make the substitution u=sin(x) If the power of sine is odd make the substitution u=cos(x) If both the powers of sine and cosine are even , keep using the formulas for the half angles to reduce to one of the previous 2 cases cos a ( x )sin b ( x ) dx ! how do we pick the right u?
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cos 4 ( x )sin 5 ( x ) dx ! Odd power of sine Even power of cosine Follow the recipe for when an odd power of sine shows up…
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If the power of the sine is odd: - Save one power of sine - Use the identity cos 2 (x)+sin 2 (x)=1 to express the remaining powers of sin(x) in terms of cos(x) - Now you should be left with a single sin(x), and a bunch of powers of cos(x)… - Make the substitution u=cos(x) - You should have a polynomial in u. Expand it!
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