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hw8-solutions(3) - PX1122 Homework 8 Due before 10 am on...

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Unformatted text preview: PX1122 - Homework 8: Due before 10 am on Friday, 16th December. 1. Compute the following integrals: [4] (a) Z π sin 3 x cos xdx, (b) Z π sin 3 x sin 2 xdx, (c) Z π cos 3 x cos xdx, (d) Z 1 x ln xdx. Answer: (a) Z π sin 3 x cos xdx =- 1 2 cos 2 x- 1 8 cos 4 x π = 0 . (b) Z π sin 3 x sin 2 xdx = 1 2 sin x- 1 10 sin 5 x π = 0 . (c) Z π cos 3 x cos xdx = 1 4 sin 2 x + 1 8 sin 4 x π = 0 . (d) Z 1 x ln xdx =- x 4 4 + x 2 ln x 2 1 =- 1 4 . 2. Compute the following integrals by partial fractions: [3] (a) Z 1 x 2- x- 12 ( x + 2)( x + 4) dx, (b) Z 1 2 x + 3 ( x + 2) 2 ( x + 3) dx, (c) Z 3 1 (3 x 2 + 5 x + 2) dx. Answer: (a) I = x- 3 ln( x + 2)- 4 ln( x + 4) | 1 = 1 + ln(2048 / 16875) ’ - 1 . 11 . (b) I = 1 x + 2 + 3 ln( x + 2)- 3 ln( x + 3) 1 =- 1 6 + ln 729 512 ’ = 0 . 187 . (c) I =- ln( x + 1) + ln(3 x + 2) | 3 = ln(11 / 8) ’ . 318 . 3. Compute the following integrals by parts and determine the value of the constant A if the integrals are all equal to 1: [3] (a) A Z π/ 2 ( x 2- 3 x ) sin xdx, (b) A Z ∞ ( x 3- x 2 + 1) e- x dx, (c) A Z π/ 2 sin 3 xdx. Answer: (a) I = A- ( x 2- 2 ) cos x + 2 x sin x- 3 sin x + 3 x cos x π/ 2 = π- 5 ⇒ A = 1 π- 5 . (b) I = Ae- x (- x 3- 2 x 2- 4 x- 5 ) ∞ = 5 ⇒ A = 1 5 ....
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hw8-solutions(3) - PX1122 Homework 8 Due before 10 am on...

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