L3 Trig Integrals Part I

# L3 Trig Integrals Part I - ± 1 48 sin 3 2 x c 6 ex 6(7.2#3...

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Lecture 3: Techniques for Integration(II) Trigonometric Integrals, part I Recall sin 2 x = 2 sin x cos x cos 2 x = 1 2 (1 + cos 2 x ) sin 2 x = 1 2 (1 ± cos 2 x ) ex. 1. Z sin 2 xdx (sin x; cos x : Even Power) ( 1 2 ( x ± 1 2 sin 2 x ) + c ) 1

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ex. 2. Z cos 3 xdx (sin x; cos x : Odd Power) (sin x ± sin 3 x 3 + c ) 2
ex. 3. Z sin 4 xdx ( 3 8 x ± 1 4 sin 2 x + 1 32 sin 4 x + c ) 3

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ex. 4. Z sin 5 x cos 2 xdx ( ± cos 7 x 7 + 2 5 cos 5 x ± 1 3 cos 3 x + c ) (sin x; cos x : Odd before Even) 4
Z sin m x cos n xdx (Odd before Even) (1) If m (or n ) is odd , save one sin x (or cos x ) and use the identity sin 2 x = 1 ± cos 2 x (or cos 2 x = 1 ± sin 2 x ) to express the remaining factors in terms of cos x (or sin x ). Then use u ± sub, let u = cos x (or u = sin x ) (2) If both m and n are even , use sin 2 x = 1 2 (1 ± cos(2 x )) ; cos 2 x = 1 2 (1+cos(2 x )) : Or sin x cos x = 1 2 sin(2 x ) : 5

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ex. 5.(7.2, #2). Z sin 4 x cos 2 xdx ( 1 16 x ± 1 64 sin 4 x

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Unformatted text preview: ± 1 48 sin 3 2 x + c ) 6 ex. 6.(7.2, #3). Z sin 5 x cos 3 xdx ( sin 6 x 6 ± sin 8 x 8 + c ) (odd power), 7 Recall sin A cos B = 1 2 [sin( A ± B ) + sin( A + B )] sin A sin B = 1 2 [cos( A ± B ) ± cos( A + B )] cos A cos B = 1 2 [cos( A ± B ) + cos( A + B )] ex. 7. R sin(4 x ) cos(5 x ) dx ( 1 2 cos( x ) ± 1 18 cos(9 x ) + c ) Tonight’s homework: Free Tutoring? Yes! Check it out in the course homepage. Students are responsible for problems: 7.2: 1 ² 49 8...
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L3 Trig Integrals Part I - ± 1 48 sin 3 2 x c 6 ex 6(7.2#3...

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