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Problem 3.05PP
Find the Laplace transform of the following time functions (* denotes convolution):
(a) ^ t) = sin t sin 7t
(b) fl:0 = s in 2 f+ 7 co s2 f
(c) f{f) = {sin t)A
( ^ ) / ( O = / c o s ( ^ — t ) sin rd r 
0
(̂ ) f i t ) = J cos(f - T ) sin xdx 
0
Step-by-step solution
(a)
Consider the time function
/ ( r )= s io /s in 7 r (1)
Consider the equation.
suu> /s in6 /K -^cos(|a -6 |/)— ...... (2)
Compare Equation (1) and (2) and find a and b. 
a = hb = 7
Substitute 1 for a and 7 for b in Equation (2).
/(0 = ̂ cos(|l-7|»)-icos(|l+7|f)
=-cos6f--cos8r 
2 2
Take Laplace transform.
= i [ ^ ______ f _ l2U'+36 i*+64j 
I4«
( j’ +36)(j*+64)
Thus, the Laplace transfomi is
14f
+36)^5^+64)
(b)
Consider the time function. 
/(r)=sin*/+7coŝ / (3)
Consider the trigonometric fomiula. 
. 2 . l-cos2/
cos'f *-
2
l+cos2r
(4)
(5)
Substitute Equation (4) and (5) in Equation (3).
= 4+3cos2/
Take Laplace transform.
■ KHA)
7̂ +̂ 16 
s (s“ + 4 )
Thus, the Laplace transfomi is
75*+ 16
r(j*+4)
(c)
Consider that division by time is equivalent to integration in the frequency domain
F { s )= ]e - f ( l)d l 
Jf (j)* = J je -“ / { l ) d l
Interchange the order of integration.
Consider the time function.
/ ( 0 = ^
= lan‘'(oo)-tan"'(5)
- f - t a n - ' M
Thus, the Laplace transfomi is
step 10 of 11
(d)
Consider the time function.
/( /)s$ in /* s in /..... (6)
Take Laplace transform. 
L ( s m ( ) = - jL - (7)
' ' 5 +1
Substitute Equation (7) in Equation (6). 
1
5 *+ 2 5 *+ !
Thus, the Laplace transfomi is 1
5* + 2j * + 1
Step 11 of 11
(e)
Consider the time function.
/ ( / ) = J cos(/ - r)sin rdr
0
Take Laplace transform.
cos(f-r)sinr