Posted by **Sara** on Saturday, July 27, 2013 at 9:33pm.

Use the fact that L{f'}=sF-f(0) to derive the formula for L{cosh at}

- Calculus HELP! -
**Steve**, Sunday, July 28, 2013 at 5:51am
cosh(at) = d/dt (1/a sinh(at))

sinh(at) = d/dt (1/a cosh(at))

So,

Since L{f(at)} = 1/|a| F(s/a)

L{cosh(at)} = sL{1/a sinh(t)}

= s/a L{sinh(t)}

L{sinh(at)} = sL{1/a cosh(at)}

= s/a L{cosh(at)} - s/a

So,

L{cosh(at)} = s/a (s/a L{cosh(at)} - s/a

L{cosh(at)}(1-s^2/a^2) = -s/a

= (s/a^2)/(s^2/a^2 - 1)

= s/(s^2-a^2)

Better check my algebra in the middle steps. I think I dropped an "a" somewhere, but the final answer is correct. You work this in the same way you do integration by parts twice for trig functions.

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