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Physics LibreTexts

14.3: The First Integration Theorem

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First Integration Theorem

The theorem is:

Lt0y(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.

L1ˉy(s)s=t0y(x)dx.

Example 14.3.1

Calculate

L11s(sa).

Solution

From the table, we see that L11sa=eat. The integration theorem tells us that

L11s(sa)=toeaxdx=(eat1)/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

Lt0y(x)dx=0(t0y(x)dx)estdt=1s0(t0y(x)dx)d(est)=[1sestt0y(x)dx]t=0+1s0esty(t)dt.

The expression in brackets is zero at both limits, and therefore the theorem is proved.


This page titled 14.3: The First Integration Theorem is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform.

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