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quiz2 solns

# Thomas' Calculus: Early Transcendentals

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Unformatted text preview: MAT 104 Spring 2003 Quiz 2 (Sections 7.6, 8.1, 8.3, 8.4) 1. (10 points) Find Z dx (1- x 2 ) 3 / 2 Make the substitution x = sin θ . Then dx = cos θ dθ and 1- x 2 = cos 2 θ . Z dx (1- x 2 ) 3 / 2 = Z cos θ cos 3 θ dθ = Z sec 2 θ dθ = tan θ + C = x √ 1- x 2 + C. 2. (10 points) Find Z (sin 5 x cos 3 x + cos 5 x sin 3 x ) dx Here we can recognize the addition formula for the sine function: sin(8 x ) = sin(3 x + 5 x ) = sin 3 x cos 5 x + cos 3 x sin 5 x. So we have simply Z sin(8 x ) dx =- cos 8 x 8 + C. Alternatively, if you did not recognize the addition formula for sine, you can use the formula from the text: sin α cos β = 1 2 [sin( α + β ) + sin( α- β )] Applying this with α = 5 x and β = 3 x we find sin 5 x cos 3 x = 1 2 [sin 8 x + sin 2 x ] Taking α = 3 x and β = 5 x we get sin 3 x cos 5 x = 1 2 [sin 8 x + sin(- 2 x )] = 1 2 [sin 8 x- sin 2 x ] . So sin 5 x cos 3 x + sin 3 x cos 5 x = 1 2 [2 sin 8 x ] = sin 8 x and now we can integrate to get- cos(8 x ) / 8 + C .....
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quiz2 solns - MAT 104 Spring 2003 Quiz 2(Sections 7.6 8.1...

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