Solutions_Assignment2

Solutions_Assignment2 - Section 7.2: 14) Evaluate the...

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Unformatted text preview: Section 7.2: 14) Evaluate the integral, Z cos cos 5 (sin ) d. Solution: Substitute u = sin and du = cos d to obtain, Z cos cos 5 (sin ) d = 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 cos 5 (sin ) d = 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 cos 5 (sin ) d = 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 )- 2 3 sin 3 (sin ) + 1 5 sin 5 (sin ) + C. 28) Evaluate the integral, Z tan 5 sec 3 d. Solution: Separate out a term of the form sec tan Z tan 4 sec 2 (sec tan ) d. Then use the identity tan 2 = sec 2 - 1 to get, Z (sec 2 - 1) 2 sec 2 (sec tan ) d. Next, make a substitution u = sec and du = sec tan d 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 - 2 5 sec 5 + 1 3 sec 3 + C. 1 1 2 3 4 1 0.8 0.6 0.4 0.2 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....
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This note was uploaded on 02/22/2012 for the course CS cs136 taught by Professor Cormack during the Winter '10 term at Waterloo.

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Solutions_Assignment2 - Section 7.2: 14) Evaluate the...

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