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

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

{[ snackBarMessage ]}

Page1 / 13

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online