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Quiz3WS2009

# Quiz3WS2009 - a Z π 2 sin n x dx = n-1 n Z π 2 sin n-2 x...

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Quiz 3 Instructor: Stephan Spencer 9 February, 2009 Student Name: 1. Compute the following Integrals a). Z tan 2 ( x ) sec ( x ) dx b). Z e 2 θ sin (3 θ ) c). Z t 0 e s sin ( t - s ) ds d). Z arccos x dx e). Z x 5 e x dx f). Z x (2 x + 3) 99 dx 2. A finite Fourier series is given by the sum f ( x ) = a 0 sin ( x ) + a 1 sin (2 x ) + . . . + a n sin ( nx ) where a 0 , a 1 , . . . a n are real numbers. We will try to establish a formula which generate these numbers. a). Show that Z π - π sin 2 ( mx ) dx = π where m is any integer. b). Show that Z π - π sin ( mx ) sin ( nx ) dx = 0 if m 6 = n . 1

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c). Show that a m = 1 π Z π - π f ( x ) sin ( mx ) dx 3. Show the following identities are true:
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Unformatted text preview: a). Z π/ 2 sin n x dx = n-1 n Z π/ 2 sin n-2 x dx for n ≥ 2 an integer. b). Z (ln x ) n dx = x (ln x ) n-n Z (ln x ) n-1 dx c). Z x m sin x dx =-x m cos x + m Z x m-1 cos x dx 4. Compute the following limits a). lim x →∞ x 3 e x 2 b). lim x →∞ x sin 1 x c). lim x →∞ ( e x + x ) 1 /x d). lim x → + ( 1 x-ln x ) e). lim x → ( π/ 2) + cos x 1-sin x f). lim x → 1 + ln x tan ( πx/ 2) 2...
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