https://file.aiquickdraw.com/mathbot/file/17254127174711r8m1mtd.jpg

The image contains a problem related to the Laplace transform, denoted as \( L \). The problem statement and parts of the solution seem to be about proving certain properties of the Laplace transform and then using those properties to evaluate specific functions.

### Problem Description:
Given that \( L[f(t)] = F(s) \), the problem asks to show two specific properties of the Laplace transform:

1. \( L[\sin(at)f(t)] = \frac{1}{2i} \left[F(s - ai) - F(s + ai)\right] \)
2. \( L[\cos(at)f(t)] = \frac{1}{2} \left[F(s - ai) + F(s + ai)\right] \)

Then, the problem asks to evaluate:

(i) \( L[\sin(2t)\sin(3t)] \)

(ii) \( L[\cos(3t)\cos(2t)] \)

### Steps to Solve:

1. **Show the properties**:
   - For \( L[\sin(at)f(t)] \), using the definition of the Laplace transform and some trigonometric identities, the proof can be derived.
   - For \( L[\cos(at)f(t)] \), a similar process is used, with the final result expressed as a sum rather than a difference.

2. **Evaluate the given Laplace Transforms**:
   - Use the identities proven in the first part to evaluate the specific Laplace transforms for the given trigonometric products.

Would you like a detailed step-by-step solution for each part of this problem?

Given that L[f(t)] = F(s), show that: 1. L[sin(at)f(t)] = (1/2i)[F(s - ai) - F(s + ai)], 2. L[cos(at)f(t)] = (1/2)[F(s - ai) + F(s + ai)], and evaluate (i) L[sin(2t)sin(3t)] (ii) L[cos(3t)cos(2t)].

Explore the properties of the Laplace Transform applied to trigonometric functions and evaluate specific Laplace transforms of products of sine and cosine functions. Suitable for undergraduate students studying engineering or mathematics.

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