# 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.

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