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The image shows a problem in Japanese, asking for the Laplace transform of the product of two given functions. The functions are:

1. \( t \) and \( e^{\lambda t} \)
2. \( e^{\lambda t} \) and \( \sin(\mu t) \)
3. \( \cos(\lambda t) \) and \( \cos(\lambda t) \)

Let’s go through the steps to solve these, assuming we need to apply the Laplace transform of the product of functions. 

### 1. Laplace Transform of \( t \cdot e^{\lambda t} \)
The Laplace transform of \( t \) is:
\[
\mathcal{L}\{t\} = \frac{1}{s^2}
\]
And the Laplace transform of \( e^{\lambda t} \) is:
\[
\mathcal{L}\{e^{\lambda t}\} = \frac{1}{s - \lambda}
\]

The Laplace transform of their product can be computed using the convolution theorem, which states:
\[
\mathcal{L}\{f(t) \cdot g(t)\} = \int_0^\infty F(s - u) G(u) \, du
\]

### 2. Laplace Transform of \( e^{\lambda t} \cdot \sin(\mu t) \)
We have:
\[
\mathcal{L}\{e^{\lambda t}\} = \frac{1}{s - \lambda}
\]
\[
\mathcal{L}\{\sin(\mu t)\} = \frac{\mu}{s^2 + \mu^2}
\]

Again, we will apply the convolution of these two functions.

### 3. Laplace Transform of \( \cos(\lambda t) \cdot \cos(\lambda t) \)
\[
\mathcal{L}\{\cos(\lambda t)\} = \frac{s}{s^2 + \lambda^2}
\]

For the product, we will again use the convolution theorem to compute the final Laplace transform.

Would you like me to proceed with detailed steps for the convolutions, or clarify any part of the process? 

### Related Questions:
1. How do you apply the convolution theorem in the Laplace domain?
2. What are the properties of Laplace transforms for trigonometric functions like \( \sin(t) \) and \( \cos(t) \)?
3. How do you handle the Laplace transform of a product of exponential and polynomial functions?
4. What are common Laplace transform pairs for reference?
5. Can we simplify Laplace transforms using known properties like linearity?

**Tip:** Using Laplace transform tables can simplify the process significantly, especially when dealing with convolutions.

求めよ：次の２つの関数の合成積とそのラプラス変換を求めよ。
1. t, e^{λt}
2. e^{λt}, sin(μt)
3. cos(λt), cos(λt)

Solve the Laplace transform of products of functions including t, e^{λt}, sin(μt), and cos(λt). This advanced problem involves using the convolution theorem and fundamental Laplace transform pairs.

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