The image contains questions on Laplace Transforms and Inverse Laplace Transforms. I will help you with the solutions and explanations for each.

Laplace Transforms:

1. The value of $L[1]$ is: $\text{Answer: } \frac{1}{s}$

2. The value of $L[a^t]$ is: $\text{Answer: } \frac{1}{s - a}$

3. The value of $L[\cos(at)]$ is: $\text{Answer: } \frac{s}{s^2 + a^2}$

4. The value of $L[e^{at}]$ is: $\text{Answer: } \frac{1}{s - a}$

5. The value of $L[t^6]$ is: $\text{Answer: } \frac{6!}{s^7} \quad \left( \text{since } L[t^n] = \frac{n!}{s^{n+1}} \right)$

6. The value of $L[e^{at}t^n]$ is: $\text{Answer: } \frac{n!}{(s - a)^{n+1}}$

7. If $L[f(t)] = F(s)$, then $L[f(at)]$ is: $\text{Answer: } \frac{1}{a}F\left( \frac{s}{a} \right)$

8. If $L[f(t)] = F(s)$ and $f(t)$ is continuous, then $L[f'(t)]$ is: $\text{Answer: } sF(s) - f(0)$

9. If $L[f(t)] = F(s)$, then $L[t f(t)]$ is: $\text{Answer: } -F'(s)$

10. If $L[f(t)] = F(s)$, then $L\left[ \int_0^t f(\tau) d\tau \right]$ is: $\text{Answer: } \frac{F(s)}{s}$

11. If $L[f(t)] = F(s)$, then $L\left[ \int_0^\infty f(t) dt \right]$ is: $\text{Answer: } \frac{F(s)}{s^2}$

Inverse Laplace Transforms:

1. The value of $L^{-1} \left[ \frac{1}{s^2 + a^2} \right]$ is: $\text{Answer: } \frac{\sin(at)}{a}$

2. The value of $L^{-1} \left[ \frac{s+a}{(s+a)^2 + 1} \right]$ is: $\text{Answer: } e^{-at} \cos(t)$

Would you like further details on any of these answers? Let me know!

Here are 5 additional related questions to practice:

1. What is the Laplace Transform of $\sin(at)$?
2. Compute the Laplace Transform of $t^n e^{bt}$.
3. Find the inverse Laplace Transform of $\frac{s}{s^2 + 4}$.
4. What is the Laplace Transform of $e^{-2t} \sin(3t)$?
5. Derive the inverse Laplace of $\frac{1}{(s-2)(s+3)}$.

Tip:

When dealing with inverse Laplace transforms involving partial fractions, always break down complex expressions into simpler fractions for easier computation.