Conflicting results for Laplace transform of sint/t!!!

[krishniv]

I stumbled upon two solutions to the equation L{sin(t)/t} and am having difficult determining which answer is correct, and why.

The two questions and solutions are below:

1. Find L^-1{-arctan(s)}.

Let f(t) = L^-1{-arctan(s)}
F(s) = -arctan(s)
F'(s) = -1/(s²+1)

Recall L^-1{F'(s)} = -t.f(t)
ie. L^-1{-1/(s²+1)} = -t.f(t)
-sin(t) = -t.f(t)
f(t) = sin(t)/t

So L^-1{-arctan(s)} = sin(t)/t.

2. Find L{sin(t)/t}.​

L{f(t)/t} = int(s->inf) F(u)du

f(t) = sin(t)
F(u) = 1/(u²+1)

L{sin(t)/t} = int(s->inf.) [1/(u²+1)]du = [arctan(u)]inf.->s = lim u->inf. arctan(u) - arctan(s) = pi/2 - arctan(s).

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Both solutions seem correct, but the first suggests L{sin(t)/t} = - arctan(s) and the second suggests L{sin(t)/t} = pi/2 - arctan(s).

If someone could help me out [ie. tell me which one is correct(or incorrect) and why] that would be awesome!! Thanks