ECE2260F09_HW5p1csoln

ECE2260F09_HW5p1csoln - s-domain 5 u 2 5 2 du s ∞ =...

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2260 F 09 HOMEWORK #5 prob 1c solution E X : Find the Laplace transform of the following waveform: sin(5 t ) t dt 0 t Hint: d dx tan 1 ( x ) = 1 x 2 + 1 S OL ' N : We start on the inside (of this layered onion) and apply identities to work our way out to the time-domain form given. The innermost term, found in a transform table, is sin(5 t ) . L sin(5 t ) { } = 5 s 2 + 5 2 Now we apply the identity for multiplication by 1/ t . L f ( t ) t = F ( u ) du s or L sin(5 t ) t = 5 u 2 + 5 2 du s Using the hint, we find the integral in a few steps. d dx tan 1 ( x / a ) = 1 ( x / a ) 2 + 1 1 a = a x 2 + a 2 Using this result, we find the integral in the
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Unformatted text preview: s-domain. 5 u 2 + 5 2 du s ∞ ∫ = tan − 1 ( u /5) u = s u = ∞ = π 2 − tan − 1 s 5 Now we apply the identity for integration in the time-domain: L f ( τ ) d τ t ∫ = F ( s ) s This implies that we need only divide our previous Laplace-domain result by s . L sin(5 t ) t dt t ∫ = 1 s π 2 − tan − 1 s 5...
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