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Unformatted text preview: rabbani (tar547) – HW07 – Radin – (56635) 1 This print-out should have 14 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Evaluate the definite integral I = integraldisplay π/ 2 (3 sin θ + sin 3 θ ) dθ . 1. I = 11 3 correct 2. I = 2 3. I = 7 3 4. I = 4 5. I = 10 3 6. I = 8 3 Explanation: Since sin 2 θ = 1- cos 2 θ we see that 3 sin θ + sin 3 θ = sin θ (3 + sin 2 θ ) = sin θ (3 + 1- cos 2 θ ) = sin θ (4- cos 2 θ ) . Thus I = integraldisplay π/ 2 sin θ (4- cos 2 θ ) dθ As the integral is now of the form sin θ f (cos θ ) , f ( x ) = 4- x 2 , the subsitution x = cos θ is suggested. For then dx =- sin θ dθ , while θ = 0 = ⇒ x = 1 , θ = π 2 = ⇒ x = 0 . In this case I =- integraldisplay 1 (4- x 2 ) dx = integraldisplay 1 (4- x 2 ) dx . Consequently, I = bracketleftBig 4 x- 1 3 x 3 bracketrightBig 1 = 11 3 . 002 10.0 points Evaluate the integral I = integraldisplay π/ 2 3 sin 3 x cos 2 x dx . 1. I = 1 5 2. I = 8 5 3. I = 6 5 4. I = 4 5 5. I = 2 5 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 . rabbani (tar547) – HW07 – Radin – (56635) 2 In this case I =- integraldisplay 1 3( u 2- u 4 ) du . Consequently, I = bracketleftBig- u 3 + 3 5 u 5 bracketrightBig 1 = 2 5 . keywords: Stewart5e, indefinite integral, powers of sin, powers of cos, trig substitu- tion, 003 10.0 points Evaluate the definite integral I = integraldisplay π/ 4 3 cos x- 4 sin x cos 3 x dx . 1. I = 1 2 2. I = 0 3. I = 3 2 4. I = 1 correct 5. I = 2 Explanation: After division 3 cos x- 4 sin x cos 3 x = 3 sec 2 x- 4 tan x sec 2 x = (3- 4 tan x ) sec 2 x . Thus I = integraldisplay π/ 4 (3- 4 tan x ) sec 2 x dx . Let u = tan x ; then du = sec 2 x dx so I = integraldisplay 1 (3- 4 u ) du = bracketleftbig 3 u- 2 u 2 bracketrightbig 1 ....
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This note was uploaded on 04/13/2010 for the course M 408L taught by Professor Radin during the Spring '08 term at University of Texas.

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solution7_pdf - rabbani (tar547) – HW07 – Radin –...

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