What is the Laplace Transform of #tcosat+sinat#?
# ℒ \ {tcosat+sinat} = (s^2-a^2)/(s^2+a^2)^2 + a/(s^2+a^2) #
Which can also be written as:
# ℒ \ {tcosat+sinat} = (s^2-a^2 + as^2+a^3)/(s^2+a^2)^2 #
We seek:
First, we observe that the linearity property of the Laplace transformation means that
Using a simple Laplace Transform lookup table, we could discover:
Thus:
I'm assuming that the above has to be derived, which is simple to do with the help of the derivative's Laplace transformation rule:
Should we allow:
When we differentiate, we obtain:
Using Laplace transforms, we obtain:
As a result, we have demonstrated;
Likewise, let's say that we now
Next, using the product rule:
Re-differentiating, we obtain:
Using Laplace transforms, we obtain:
As a result, we have demonstrated;
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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