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Math151Quiz1Sols - QUIZ 1(CHAPTER 9 SOLUTIONS MATH 151...

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QUIZ 1 (CHAPTER 9) SOLUTIONS MATH 151 – SPRING 2004 – KUNIYUKI 105 POINTS TOTAL, BUT 100 POINTS = 100% For 1) through 8), evaluate the integrals. 1) e x dx x 2 3 sin ( ) Ú (14 points) METHOD 1 Alternating signs Successive derivatives Successive antiderivatives + sin 3 x ( ) e x 2 (strike) - 3 3 cos x ( ) 1 2 2 e x + - ( ) 9 3 sin x 1 4 2 e x Ú e x dx e x e x e x dx I e x e x I I e x e x x x x x x x x x I I 2 2 2 2 2 2 2 2 3 1 2 3 3 4 3 9 4 3 1 2 3 3 4 3 9 4 13 4 1 2 3 3 4 3 sin sin cos sin sin cos sin cos " " ( ) = ( ) - ( ) - ( ) = ( ) - ( ) - = ( ) - ( ) Ú Ú Call 1 2 44 3 44 1 2 44 3 44 + = ( ) - ( ) È Î Í ˘ ˚ ˙ + = ( ) - ( ) + C I e x e x C I x x 4 13 1 2 3 3 4 3 2 2 sin cos 2 13 sin 3 3 13 cos 3 2 2 e x e x C x x
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METHOD 2 Alternating signs Successive derivatives Successive antiderivatives + e x 2 sin 3 x ( ) (strike) - 2 2 e x - ( ) È Î Í ˘ ˚ ˙ 1 3 3 cos x + 4 2 e x - ( ) È Î Í ˘ ˚ ˙ 1 9 3 sin x Ú e x dx e x e x e x dx I e x e x I I e x e x x x x x x x x I I 2 2 2 2 2 2 2 2 3 1 3 3 2 9 3 4 9 3 1 3 3 2 9 3 4 9 13 9 1 3 3 2 9 3 sin cos sin sin cos sin cos sin " " ( ) = - ( ) + ( ) - ( ) = - ( ) + ( ) - = - ( ) + Ú Ú Call 1 2 44 3 44 1 2 44 3 44 x C I e x e x C I x x ( ) + = - ( ) + ( ) È Î Í ˘ ˚ ˙ + = - ( ) + ( ) + 9 13 1 3 3 2 9 3 2 2 cos sin 3 13 cos 3 2 13 sin 3 2 2 e x e x C x x (Note that this is equivalent to our previous result.) 2) cot csc 6 4 x x dx Ú (10 points) We like the fact that the csc power is even.
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