Let's find the Laplace transform of each function step by step.

(a) $f(t) = t^3 - t^2 + 7t - 10$

The Laplace transform of $f(t)$ is given by: $\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}.$ For $f(t) = t^3 - t^2 + 7t - 10$, we compute:

1. $\mathcal{L}\{t^3\} = \frac{3!}{s^{4}} = \frac{6}{s^4}$
2. $\mathcal{L}\{-t^2\} = -\frac{2!}{s^{3}} = -\frac{2}{s^3}$
3. $\mathcal{L}\{7t\} = 7\cdot\mathcal{L}\{t\} = 7\cdot\frac{1!}{s^2} = \frac{7}{s^2}$
4. $\mathcal{L}\{-10\} = -10\cdot\mathcal{L}\{1\} = -10\cdot\frac{1}{s} = -\frac{10}{s}$

Adding these together: $\mathcal{L}\{f(t)\} = \frac{6}{s^4} - \frac{2}{s^3} + \frac{7}{s^2} - \frac{10}{s}.$

(b) $f(t) = 7t + 6e^{-t} + \sin t$

We compute the Laplace transform for each term:

1. $\mathcal{L}\{7t\} = 7\cdot\frac{1}{s^2} = \frac{7}{s^2}$
2. $\mathcal{L}\{6e^{-t}\} = 6\cdot\frac{1}{s+1} = \frac{6}{s+1}$
3. $\mathcal{L}\{\sin t\} = \frac{1}{s^2 + 1}$

Adding these together: $\mathcal{L}\{f(t)\} = \frac{7}{s^2} + \frac{6}{s+1} + \frac{1}{s^2 + 1}.$

(c) $f(t) = t e^{at} \sin(bt)$

Using the general Laplace transform formula: $\mathcal{L}\{t e^{at} \sin(bt)\} = \frac{2b(s-a)}{\left[(s-a)^2 + b^2\right]^2}.$

Substituting $a$ and $b$, the formula remains in terms of these parameters: $\mathcal{L}\{f(t)\} = \frac{2b(s-a)}{\left[(s-a)^2 + b^2\right]^2}.$

Summary of Results

1. (a) $\mathcal{L}\{f(t)\} = \frac{6}{s^4} - \frac{2}{s^3} + \frac{7}{s^2} - \frac{10}{s}$
2. (b) $\mathcal{L}\{f(t)\} = \frac{7}{s^2} + \frac{6}{s+1} + \frac{1}{s^2 + 1}$
3. (c) $\mathcal{L}\{f(t)\} = \frac{2b(s-a)}{\left[(s-a)^2 + b^2\right]^2}$

Would you like a detailed explanation for any of these steps or transformations?

Related Questions:

1. What is the Laplace transform of $t^n e^{at}$?
2. How do we handle Laplace transforms for periodic functions?
3. Can Laplace transforms be used to solve differential equations?
4. What are the initial and final value theorems in Laplace transforms?
5. How can partial fraction decomposition help simplify inverse Laplace transforms?

Tip:

Always verify the regions of convergence for the Laplace transform to ensure its validity.