Part 2 of the transform of the sin(at) | Laplace transform | Differential Equations | Khan Academy

TL;DR

The Laplace transform of sine of at is a over s squared, plus a squared, and it can be used in integration by parts problems.

Transcript

Welcome back. We were in the midst of figuring out the Laplace transform of sine of at when I was running out of time. This was the definition of the Laplace transform of sine of at. I said that also equals y. This is going to be useful for us, since we're going to be doing integration by parts twice. So I did integration by parts once, then I did ... Read More

Key Insights

• 🥳 The Laplace transform of sine of at can be derived through integration by parts.
• 🥳 Careful attention must be paid to avoid careless mistakes when performing integration by parts multiple times.
• ❎ The Laplace transform of sine of at is a over s squared, plus a squared.
• 🍉 Evaluating the boundaries of the integral involves taking the limits of the exponential and trigonometric terms.

Questions & Answers

Q: What is the Laplace transform of sine of at and how is it derived?

The Laplace transform of sine of at is a over s squared, plus a squared. It is derived by performing integration by parts twice on the original function.

Q: Why did the speaker not worry about the boundaries of the integral initially?

The boundaries of the integral were not initially considered because the focus was on solving the indefinite integral first. Once the indefinite integral was solved, the boundaries could be evaluated separately.

Q: How are the boundaries of the integral evaluated?

The boundaries are evaluated by taking the limit as t approaches infinity and as t approaches 0. For t approaching infinity, the exponential term becomes zero, and for t approaching 0, the exponential term becomes one.

Q: What is the significance of the Laplace transform of sine of at?

The Laplace transform of sine of at is an important entry in the table of Laplace transforms. It allows for the transformation of sine functions with a scaling factor in the time domain to the frequency domain.

Summary & Key Takeaways

• The video discusses the Laplace transform of sine of at and its derivation through integration by parts.

• The indefinite integral of the Laplace transform is solved, and the boundaries are evaluated.

• Careless mistakes in the calculations are identified and corrected.