M408L_HW08

# M408L_HW08 - hyun(hh7953 – HW08 – gogolev –(57440 1...

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Unformatted text preview: hyun (hh7953) – HW08 – gogolev – (57440) 1 This print-out should have 23 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Evaluate the integral I = integraldisplay π/ 2 sin 3 x cos 2 x dx . 1. I = 2 5 2. I = 1 15 3. I = 8 15 4. I = 4 15 5. I = 2 15 correct Explanation: Since sin 3 x cos 2 x = sin x (sin 2 x cos 2 x ) = sin x (1- cos 2 x )cos 2 x = sin x (cos 2 x- cos 4 x ) , the integrand is of the form sin xf (cos x ), sug- gesting use of the substitution u = cos x . For then du =- sin x dx , while x = 0 = ⇒ u = 1 x = π 2 = ⇒ u = 0 . In this case I =- integraldisplay 1 ( u 2- u 4 ) du . Consequently, I = bracketleftBig- 1 3 u 3 + 1 5 u 5 bracketrightBig 1 = 2 15 . keywords: Stewart5e, indefinite integral, powers of sin, powers of cos, trig substitu- tion, 002 10.0 points Evaluate the definite integral I = integraldisplay π/ 4 cos x- 3 sin x cos 3 x dx . 1. I = 1 2 2. I =- 1 2 correct 3. I = 1 4. I = 3 2 5. I = 0 Explanation: After division cos x- 3 sin x cos 3 x = sec 2 x- 3 tan x sec 2 x = (1- 3 tan x ) sec 2 x . Thus I = integraldisplay π/ 4 (1- 3 tan x ) sec 2 x dx . Let u = tan x ; then du = sec 2 x dx so I = integraldisplay 1 (1- 3 u ) du = bracketleftbigg u- 3 2 u 2 bracketrightbigg 1 . hyun (hh7953) – HW08 – gogolev – (57440) 2 Consequently, I =- 1 2 . 003 10.0 points Find the value of I = integraldisplay π 3 tan 4 x dx . 1. I = π 3 correct 2. I = π √ 3 3 3. I = π 6 + 8 √ 3 9 4. I = π 4 + 2 3 5. I = π 4- 2 3 6. I = π 6- 8 √ 3 9 Explanation: Since tan 2 x = sec 2 x- 1 , we see that tan 4 x = tan 2 x ( sec 2 x- 1 ) = tan 2 x sec 2 x- tan 2 x . Thus by trig identities yet again, tan 4 x = ( tan 2 x- 1 ) sec 2 x + 1 . In this case, I = integraldisplay π 3 bracketleftbig( tan 2 x- 1 ) sec 2 x + 1 bracketrightbig dx = bracketleftbigg 1 3 tan 3 x- tan x + x bracketrightbigg π 3 . On the other hand, tan π 3 = √ 3 . Consequently, I = π 3 . 004 10.0 points Evaluate the definite integral I = integraldisplay π x (3 cos 2 x- sin 2 x ) dx 1. I = π 2 + 1 2. I = π 2 3. I = 3 2 π 2 4. I = 1 2 π 2 + 1 2 5. I = 1 2 π 2 correct Explanation: Since cos 2 x = 1 2 (1 + cos 2 x ) and sin 2 x = 1 2 (1- cos 2 x ) , we see that I = 1 2 integraldisplay π x { 3 (1 + cos 2 x )- 1 + cos 2 x } dx = integraldisplay π x dx + 2 integraldisplay π x cos 2 x dx = bracketleftBig 1 2 x 2 bracketrightBig π + 2 integraldisplay π x cos 2 x dx = 1 2 π 2 + 2 integraldisplay π x cos 2 x dx . hyun (hh7953) – HW08 – gogolev – (57440) 3 But after integration by parts, integraldisplay π x cos 2 x dx = 1 2 bracketleftBig x sin 2 x bracketrightBig π- 1 2 integraldisplay π sin 2 x dx = 0 + 1 4 bracketleftBig cos 2 x bracketrightBig π = 0 ....
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M408L_HW08 - hyun(hh7953 – HW08 – gogolev –(57440 1...

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