ℒ[cos(ωt)] = s/(s² + ω²) | LAPLACE TRANSFORM of COSINE Requires Integration by Parts

Learn how to compute the Laplace transform of cos(ωt) directly from the definition using an improper integral. This short video shows the key antiderivative you get after applying integration by parts, and then evaluates the result to show why ℒ[cos(ωt)] = s/(s² + ω²). This formula is one of the most important in Laplace transform tables — and it’s derived from scratch, using fundamental calculus ideas. Topics include: 1) Laplace transform definition and improper integral setup, 2) Integration by parts (conceptually), 3) antiderivative of e^(-s*t)*cos(ω*t), 4) evaluation at infinity and 0 when s is positive, and 5) a note that the graph of s/(s² + ω²) rises before it falls (increases before it decreases). 📖 Infinite Powers, How Calculus Reveals the Secrets of the Universe: https://amzn.to/37PBMjb. #LaplaceTransform #MathShorts #EngineeringMath #Calculus #ComplexAnalysis #MathEducation #LearnMath #MathVisualization #SignalsAndSystems #DifferentialEquations Links and resources =============================== 🔴 Subscribe to Bill Kinney Math: https://www.youtube.com/user/billkinn... 🔴 Subscribe to my Math Blog, Infinity is Really Big: https://infinityisreallybig.com/ 🔴 Follow me on Twitter:   / billkinneymath   🔴 Follow me on Instagram:   / billkinneymath   🔴 You can support me by buying "Infinite Powers, How Calculus Reveals the Secrets of the Universe", by Steven Strogatz, or anything else you want to buy, starting from this link: https://amzn.to/3eXEmuA. 🔴 Check out my artist son Tyler Kinney's website: https://www.tylertkinney.co/ 🔴 Desiring God website: https://www.desiringgod.org/ AMAZON ASSOCIATE As an Amazon Associate I earn from qualifying purchases.