14.3: The First Integration Theorem
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First Integration Theorem
The theorem is:
L∫t0y(x)dx=ˉy(s)s.
Before deriving this theorem, here's a quick example to show what it means. The theorem is most useful, as in this example, for finding an inverse Laplace transform, i.e.
L−1ˉy(s)s=∫t0y(x)dx.
Example 14.3.1
Calculate
L−11s(s−a).
Solution
From the table, we see that L−11s−a=eat. The integration theorem tells us that
L−11s(s−a)=∫toeaxdx=(eat−1)/a.
You should now verify that this is the correct answer by substituting this in Equation 14.1.2 and integrating – or (and!) using the table of Laplace transforms.
Proof
The proof of the theorem is just a matter of integrating by parts. Thus
L∫t0y(x)dx=∫∞0(∫t0y(x)dx)e−stdt=−1s∫∞0(∫t0y(x)dx)d(e−st)=[−1se−st∫t0y(x)dx]∞t=0+1s∫∞0e−sty(t)dt.
The expression in brackets is zero at both limits, and therefore the theorem is proved.