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resolver cada ejercicio hasta llegar al resultado

To solve the Laplace transforms presented in the image, I'll walk through each exercise step by step.

### (a) \( \mathcal{L}[(2t + 1)^3 - 8t^3] \)
To solve this, we use the linearity of the Laplace transform:

1. Expand \( (2t + 1)^3 \) using the binomial theorem:
   \[
   (2t + 1)^3 = 8t^3 + 12t^2 + 6t + 1
   \]
   
2. Apply the Laplace transform to each term:
   \[
   \mathcal{L}[8t^3] = \frac{8 \cdot 3!}{s^4} = \frac{48}{s^4}
   \]
   \[
   \mathcal{L}[12t^2] = \frac{12 \cdot 2!}{s^3} = \frac{24}{s^3}
   \]
   \[
   \mathcal{L}[6t] = \frac{6 \cdot 1!}{s^2} = \frac{6}{s^2}
   \]
   \[
   \mathcal{L}[1] = \frac{1}{s}
   \]

3. Combine and subtract the terms:
   \[
   \mathcal{L}[(2t + 1)^3 - 8t^3] = \left(\frac{48}{s^4} + \frac{24}{s^3} + \frac{6}{s^2} + \frac{1}{s}\right) - \frac{48}{s^4} = \frac{24}{s^3} + \frac{6}{s^2} + \frac{1}{s}
   \]

### (b) \( \mathcal{L}\left[\sqrt{2} \sin\left(\frac{t}{2}\right)\right] \)

For a sine function:
\[
\mathcal{L}[\sin(at)] = \frac{a}{s^2 + a^2}
\]

Here, \( a = \frac{1}{2} \):
\[
\mathcal{L}\left[\sin\left(\frac{t}{2}\right)\right] = \frac{\frac{1}{2}}{s^2 + \left(\frac{1}{2}\right)^2} = \frac{1}{2\left(s^2 + \frac{1}{4}\right)}
\]

Multiplying by \( \sqrt{2} \):
\[
\mathcal{L}\left[\sqrt{2} \sin\left(\frac{t}{2}\right)\right] = \frac{\sqrt{2}}{2\left(s^2 + \frac{1}{4}\right)} = \frac{\sqrt{2}}{2s^2 + \frac{1}{2}}
\]

### (c) \( \mathcal{L}\left[e^{-at} \cdot 2 \cosh(6t)\right] \)

The hyperbolic cosine function is given by:
\[
\cosh(bt) = \frac{e^{bt} + e^{-bt}}{2}
\]

1. Use the property \( \mathcal{L}[e^{-at} f(t)] = F(s + a) \) where \( F(s) \) is the Laplace transform of \( f(t) \).

2. For \( \mathcal{L}[\cosh(6t)] \), recognize that:
   \[
   \mathcal{L}[\cosh(6t)] = \frac{s}{s^2 - 36}
   \]
   Multiply by 2:
   \[
   2 \cdot \mathcal{L}[\cosh(6t)] = \frac{2s}{s^2 - 36}
   \]

3. Applying the shift:
   \[
   \mathcal{L}[e^{-at} \cdot 2 \cosh(6t)] = \frac{2(s+a)}{(s+a)^2 - 36}
   \]

### (d) \( 2 \mathcal{L}\left[ e

Explore how to solve complex Laplace transforms step-by-step with detailed examples. This page covers topics such as the linearity of Laplace transforms, Laplace transforms of trigonometric functions, and applications of the shift theorem. Suitable for advanced college students and professionals.

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