How to Calculate L{∫₀ᵗ f(τ)dτ}: Step-by-Step Guide for Integrals

📚 Understanding Laplace Transforms of Integrals

The Laplace transform is a powerful tool for solving differential equations and analyzing linear systems. It transforms a function of time, $f(t)$, into a function of complex frequency, $F(s)$. When dealing with integrals within the Laplace transform, a specific formula can significantly simplify the process. Let's explore the step-by-step guide to calculate $L\{\int_0^t f(\tau)d\tau\}$.

📜 Historical Background

The Laplace transform is named after Pierre-Simon Laplace, who introduced it in his work on probability theory. However, the integral transform itself was first used by Leonhard Euler. Over time, mathematicians and engineers developed the theory and applications of the Laplace transform, making it an indispensable tool in various fields.

🔑 Key Principles

• 🔍 Definition of Laplace Transform: The Laplace transform of a function $f(t)$ is defined as $F(s) = L\{f(t)\} = \int_0^{\infty} e^{-st} f(t) dt$.
• 💡 Laplace Transform of an Integral: The key principle is that $L\{\int_0^t f(\tau)d\tau\} = \frac{F(s)}{s}$, where $F(s)$ is the Laplace transform of $f(t)$.
• 📝 Linearity: The Laplace transform is a linear operator, meaning that $L\{af(t) + bg(t)\} = aL\{f(t)\} + bL\{g(t)\} = aF(s) + bG(s)$, where $a$ and $b$ are constants.
• ⏱️ Time Invariance: This property is more relevant for signals but it is good to know, because the Laplace transform can be used in signal processing.

✍️ Step-by-Step Calculation

1. ✔️ Identify $f(t)$: Determine the function $f(t)$ that is being integrated within the Laplace transform.
2. 🧮 Find $F(s)$: Calculate the Laplace transform of $f(t)$, denoted as $F(s) = L\{f(t)\}$. You can use Laplace transform tables or direct integration.
3. ➗ Divide by $s$: Divide the Laplace transform $F(s)$ by $s$ to obtain the Laplace transform of the integral. That is, $L\{\int_0^t f(\tau)d\tau\} = \frac{F(s)}{s}$.

🧪 Real-world Examples

Example 1:

Let $f(t) = t$. Find $L\{\int_0^t \tau d\tau\}$.

1. ✔️ Identify $f(t)$: $f(t) = t$.
2. 🧮 Find $F(s)$: $F(s) = L\{t\} = \frac{1}{s^2}$.
3. ➗ Divide by $s$: $L\{\int_0^t \tau d\tau\} = \frac{1}{s^2} / s = \frac{1}{s^3}$.

Example 2:

Let $f(t) = e^{-at}$. Find $L\{\int_0^t e^{-a\tau} d\tau\}$.

1. ✔️ Identify $f(t)$: $f(t) = e^{-at}$.
2. 🧮 Find $F(s)$: $F(s) = L\{e^{-at}\} = \frac{1}{s+a}$.
3. ➗ Divide by $s$: $L\{\int_0^t e^{-a\tau} d\tau\} = \frac{1}{s(s+a)}$.

✍️ Advanced Examples

Example 3:

Let $f(t) = sin(t)$. Find $L\{\int_0^t sin(\tau) d\tau\}$.

1. ✔️ Identify $f(t)$: $f(t) = sin(t)$.
2. 🧮 Find $F(s)$: $F(s) = L\{sin(t)\} = \frac{1}{s^2 + 1}$.
3. ➗ Divide by $s$: $L\{\int_0^t sin(\tau) d\tau\} = \frac{1}{s(s^2 + 1)}$.

Example 4:

Let $f(t) = cos(t)$. Find $L\{\int_0^t cos(\tau) d\tau\}$.

1. ✔️ Identify $f(t)$: $f(t) = cos(t)$.
2. 🧮 Find $F(s)$: $F(s) = L\{cos(t)\} = \frac{s}{s^2 + 1}$.
3. ➗ Divide by $s$: $L\{\int_0^t cos(\tau) d\tau\} = \frac{s}{s(s^2 + 1)} = \frac{1}{s^2 + 1}$.

🤔 Conclusion

Calculating the Laplace transform of an integral simplifies to finding the Laplace transform of the integrand and dividing by $s$. This technique is valuable in solving integral equations and analyzing systems involving integrated functions.

📝 Practice Quiz

Solve the following Laplace transform problems:

1. ❓ Find $L\{\int_0^t \tau^2 d\tau\}$.
2. ❓ Find $L\{\int_0^t cos(2\tau) d\tau\}$.
3. ❓ Find $L\{\int_0^t e^{3\tau} d\tau\}$.
4. ❓ Find $L\{\int_0^t (\tau + 1) d\tau\}$.
5. ❓ Find $L\{\int_0^t sin(2\tau) d\tau\}$.
6. ❓ Find $L\{\int_0^t (e^{-\tau} + 1) d\tau\}$.
7. ❓ Find $L\{\int_0^t \tau e^{-\tau} d\tau\}$.