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Solutions_Assignment2

Solutions_Assignment2 - Section 7.2 14 Evaluate the...

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Section 7.2: 14) Evaluate the integral, Z cos θ cos 5 (sin θ ) dθ. Solution: Substitute u = sin θ and du = cos θdθ to obtain, Z cos θ cos 5 (sin θ ) = Z cos 5 u du Next, we separate one of the cosines and use the trigonometric identity that cos 2 x = 1 - sin 2 x , Z cos θ cos 5 (sin θ ) = Z (1 - sin 2 u ) 2 cos u du = Z (1 - 2 sin 2 u + sin 4 u ) cos u du Make another substitution w = sin u and dw = cos udu to finally obtain, Z cos θ cos 5 (sin θ ) = Z (1 - 2 w 2 + w 4 ) dw, = w - 2 3 w 3 + 1 5 w 5 + C, = sin u - 2 3 sin 3 u + 1 5 sin 5 u + C, = sin(sin θ ) - 2 3 sin 3 (sin θ ) + 1 5 sin 5 (sin θ ) + C. 28) Evaluate the integral, Z tan 5 θ sec 3 θ dθ. Solution: Separate out a term of the form sec θ tan θ Z tan 4 θ sec 2 θ (sec θ tan θ ) dθ. Then use the identity tan 2 θ = sec 2 θ - 1 to get, Z (sec 2 θ - 1) 2 sec 2 θ (sec θ tan θ ) dθ. Next, make a substitution u = sec θ and du = sec θ tan θdθ to get and integral that we can evaluate after expanding Z ( u 2 - 1) 2 u 2 du = Z ( u 6 - 2 u 4 + u 2 ) du, = 1 7 u 7 - 2 5 u 5 + 1 3 u 3 + C, = 1 7 sec 7 θ - 2 5 sec 5 θ + 1 3 sec 3 θ + C. 1
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0 1 2 3 4 ï 1 ï 0.8 ï 0.6 ï 0.4 ï 0.2 0 0.2 0.4 0.6 0.8 1 Interior sin 3 x cos 3 x x = / /4 x = 5 / /4 58) Find the area of the region bounded by the given curves. y = sin 3 x, y = cos 3 x, π/ 4 x 5 π/ 4 .
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