Detailed examples: Laplace Transform derivation for sin(bt), cos(bt), and t^n.

📚 Quick Study Guide

• 🔍 The Laplace Transform converts a function of time, $f(t)$, into a function of complex frequency, $F(s)$. It's defined as: $F(s) = \int_0^{\infty} e^{-st} f(t) dt$.
• 🔢 Laplace Transform of $sin(bt)$: $L{sin(bt)} = \frac{b}{s^2 + b^2}$
• 📈 Laplace Transform of $cos(bt)$: $L{cos(bt)} = \frac{s}{s^2 + b^2}$
• 💡 Laplace Transform of $t^n$ (where n > -1): $L{t^n} = \frac{n!}{s^{n+1}}$ for integer $n \geq 0$. If $n$ is not an integer $L{t^n} = \frac{\Gamma(n+1)}{s^{n+1}}$ where $\Gamma$ is the Gamma function.
• 📝 Remember that $s$ is a complex number and the real part of $s$ must be greater than 0 for the integral to converge.

🧪 Practice Quiz

1. Question 1: What is the Laplace Transform of $sin(2t)$?
   1. $\frac{s}{s^2 + 4}$
   2. $\frac{2}{s^2 + 4}$
   3. $\frac{4}{s^2 + 2}$
   4. $\frac{s}{s^2 + 2}$
2. Question 2: What is the Laplace Transform of $cos(3t)$?
   1. $\frac{3}{s^2 + 9}$
   2. $\frac{s}{s^2 + 9}$
   3. $\frac{9}{s^2 + 3}$
   4. $\frac{s}{s^2 + 3}$
3. Question 3: What is the Laplace Transform of $t^2$?
   1. $\frac{1}{s^3}$
   2. $\frac{2}{s^3}$
   3. $\frac{1}{s^2}$
   4. $\frac{2}{s^2}$
4. Question 4: What is the Laplace Transform of $sin(5t)$?
   1. $\frac{s}{s^2 + 25}$
   2. $\frac{5}{s^2 + 25}$
   3. $\frac{25}{s^2 + 5}$
   4. $\frac{s}{s^2 + 5}$
5. Question 5: What is the Laplace Transform of $cos(4t)$?
   1. $\frac{4}{s^2 + 16}$
   2. $\frac{s}{s^2 + 16}$
   3. $\frac{16}{s^2 + 4}$
   4. $\frac{s}{s^2 + 4}$
6. Question 6: What is the Laplace Transform of $t^3$?
   1. $\frac{3}{s^4}$
   2. $\frac{6}{s^4}$
   3. $\frac{3}{s^3}$
   4. $\frac{6}{s^3}$
7. Question 7: What is the Laplace Transform of $sin(at)$?
   1. $\frac{s}{s^2 + a^2}$
   2. $\frac{a}{s^2 + a^2}$
   3. $\frac{a^2}{s^2 + a}$
   4. $\frac{s}{s^2 + a}$