To find the Laplace transform of the function \(\frac{1 - \cos t}{t}\), we can use the following method:

### Step 1: Identify the function
The given function is \(\frac{1 - \cos t}{t}\).

### Step 2: Use the known Laplace transform
We know that the Laplace transform of the function \(\text{Si}(t) = \int_0^t \frac{\sin u}{u} \, du\) is \(\mathcal{L}\{\text{Si}(t)\} = \frac{\tan^{-1}(s)}{s}\).

### Step 3: Express the given function in terms of a known transform
The function \(\frac{1 - \cos t}{t}\) can be recognized as the derivative of the sine integral function \(\text{Si}(t)\):
\[
\frac{d}{dt} \text{Si}(t) = \frac{\sin t}{t} \implies \int_0^t \frac{\sin u}{u} \, du = \text{Si}(t)
\]
Thus,
\[
\frac{1 - \cos t}{t} = \int_0^t \frac{\sin u}{u} \, du = \text{Si}(t)
\]
Therefore, the Laplace transform of \(\frac{1 - \cos t}{t}\) is the same as the Laplace transform of \(\text{Si}(t)\).

### Step 4: Find the Laplace transform
\[
\mathcal{L}\left\{\frac{1 - \cos t}{t}\right\} = \frac{\tan^{-1}(s)}{s}
\]

So, the Laplace transform of \(\frac{1 - \cos t}{t}\) is \(\frac{\tan^{-1}(s)}{s}\).

Would you like further details on this process, or have any questions?

Here are 5 related questions:
1. What is the Laplace transform of \(\frac{\sin t}{t}\)?
2. How do you find the inverse Laplace transform of \(\frac{\tan^{-1}(s)}{s}\)?
3. What are the applications of the Laplace transform in solving differential equations?
4. How do you derive the Laplace transform of \(\sin(at)\)?
5. What is the Laplace transform of a Heaviside step function?

**Tip:** The Laplace transform is particularly useful for solving linear ordinary differential equations, as it converts differential equations into algebraic equations that are easier to manipulate.

Discover how to find the Laplace transform of (1 - cos t)/t step-by-step with detailed explanations and formulas. Learn about integral transforms and special functions. Suitable for advanced undergraduate students and those interested in mathematical analysis.

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