3. Calculate the following integral [you have to show your manual calculations leading to the answer although you are allowed to use a math tool to check your answer): S6(2t-5) Pei +2t +3 dt
3. Calculate the following integral [you have to show your manual calculations leading to the answer although you are allowed to use a math tool to check your answer): S6(2t-5) Pei +2t +3 dt
Introductory Circuit Analysis (13th Edition)
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ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
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**Problem 3.**
**Calculate the following integral [you have to show your manual calculations leading to the answer although you are allowed to use a math tool to check your answer]:**
\[
\int_{-\infty}^{\infty} \delta(2t - 5) \left( t^2 e^{-\frac{t+5}{2}} + 2t + 3 \right) \, dt
\]
---
**Explanation:**
This is an integral involving the Dirac delta function, \(\delta(2t - 5)\), and the expression \(t^2 e^{-\frac{t+5}{2}} + 2t + 3\). The Dirac delta function is a special mathematical function that is very useful in various fields, particularly in physics and engineering, because it “picks out” the value of another function at a specific point.
- **Dirac Delta Function:** \(\delta(2t - 5)\) implies that the integral will evaluate the expression only at the point where \(2t - 5 = 0\), which simplifies to \(t = \frac{5}{2}\).
- **Expression within the integral:** The expression inside the integral is \(t^2 e^{-\frac{t+5}{2}} + 2t + 3\), and it will be evaluated at \(t = \frac{5}{2}\).
- **Calculation Step:** Substitute \(t = \frac{5}{2}\) into the expression:
\[ \left( \left(\frac{5}{2}\right)^2 e^{-\frac{\frac{5}{2}+5}{2}} + 2\left(\frac{5}{2}\right) + 3 \right) \]
This simplification and evaluation are critical in understanding the behavior of the integral with the delta function.
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Transcribed Image Text:Certainly! Here is the transcription of the image for an educational website:
---
**Problem 3.**
**Calculate the following integral [you have to show your manual calculations leading to the answer although you are allowed to use a math tool to check your answer]:**
\[
\int_{-\infty}^{\infty} \delta(2t - 5) \left( t^2 e^{-\frac{t+5}{2}} + 2t + 3 \right) \, dt
\]
---
**Explanation:**
This is an integral involving the Dirac delta function, \(\delta(2t - 5)\), and the expression \(t^2 e^{-\frac{t+5}{2}} + 2t + 3\). The Dirac delta function is a special mathematical function that is very useful in various fields, particularly in physics and engineering, because it “picks out” the value of another function at a specific point.
- **Dirac Delta Function:** \(\delta(2t - 5)\) implies that the integral will evaluate the expression only at the point where \(2t - 5 = 0\), which simplifies to \(t = \frac{5}{2}\).
- **Expression within the integral:** The expression inside the integral is \(t^2 e^{-\frac{t+5}{2}} + 2t + 3\), and it will be evaluated at \(t = \frac{5}{2}\).
- **Calculation Step:** Substitute \(t = \frac{5}{2}\) into the expression:
\[ \left( \left(\frac{5}{2}\right)^2 e^{-\frac{\frac{5}{2}+5}{2}} + 2\left(\frac{5}{2}\right) + 3 \right) \]
This simplification and evaluation are critical in understanding the behavior of the integral with the delta function.
---
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