(a) L² (3x². 2 (3x² - 2x - 1) 8 (x - 3)dx (b) *(cos x) 8(x − n)dx 5 (c)( (cos x)8(x - 2π)dx
(a) L² (3x². 2 (3x² - 2x - 1) 8 (x - 3)dx (b) *(cos x) 8(x − n)dx 5 (c)( (cos x)8(x - 2π)dx
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Transcribed Image Text:This problem is designed to give you practice using the Dirac delta function. Evaluate the following integrals. Show your reasoning.
![The image contains three mathematical expressions involving integrals with a Dirac delta function. The expressions are:
(a) \(\int_{2}^{6} (3x^2 - 2x - 1) \delta(x - 3) \, dx\)
(b) \(\int_{0}^{5} (\cos x) \delta(x - \pi) \, dx\)
(c) \(\int_{0}^{5} (\cos x) \delta(x - 2\pi) \, dx\)
### Explanation:
1. **Integrals**: Each expression is an integral that evaluates a function over a specified interval. The integrals incorporate a Dirac delta function, which is a highly peaked function at a specific point.
2. **Dirac Delta Function (\(\delta(x - a)\))**:
- Acts like a "sampling" function that picks out the value of the function at \(x = a\).
- This means that the integral of \(f(x) \delta(x - a)\) over a range that includes \(x = a\) will be \(f(a)\).
### Evaluation of Each Integral:
- **Expression (a)**:
- \(\int_{2}^{6} (3x^2 - 2x - 1) \delta(x - 3) \, dx\)
- The integral evaluates the polynomial \(3x^2 - 2x - 1\) at \(x = 3\).
- **Expression (b)**:
- \(\int_{0}^{5} (\cos x) \delta(x - \pi) \, dx\)
- The integral evaluates \(\cos x\) at \(x = \pi\).
- **Expression (c)**:
- \(\int_{0}^{5} (\cos x) \delta(x - 2\pi) \, dx\)
- The integral evaluates \(\cos x\) at \(x = 2\pi\), but since \(2\pi\) is outside the integration range [0, 5], the integral evaluates to zero.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F29c74d06-0f3b-4eb2-9c9d-dbbc1918002c%2F46e55fbf-9d5e-414e-9d96-b06603ce0a65%2F8o6muio_processed.png&w=3840&q=75)
Transcribed Image Text:The image contains three mathematical expressions involving integrals with a Dirac delta function. The expressions are:
(a) \(\int_{2}^{6} (3x^2 - 2x - 1) \delta(x - 3) \, dx\)
(b) \(\int_{0}^{5} (\cos x) \delta(x - \pi) \, dx\)
(c) \(\int_{0}^{5} (\cos x) \delta(x - 2\pi) \, dx\)
### Explanation:
1. **Integrals**: Each expression is an integral that evaluates a function over a specified interval. The integrals incorporate a Dirac delta function, which is a highly peaked function at a specific point.
2. **Dirac Delta Function (\(\delta(x - a)\))**:
- Acts like a "sampling" function that picks out the value of the function at \(x = a\).
- This means that the integral of \(f(x) \delta(x - a)\) over a range that includes \(x = a\) will be \(f(a)\).
### Evaluation of Each Integral:
- **Expression (a)**:
- \(\int_{2}^{6} (3x^2 - 2x - 1) \delta(x - 3) \, dx\)
- The integral evaluates the polynomial \(3x^2 - 2x - 1\) at \(x = 3\).
- **Expression (b)**:
- \(\int_{0}^{5} (\cos x) \delta(x - \pi) \, dx\)
- The integral evaluates \(\cos x\) at \(x = \pi\).
- **Expression (c)**:
- \(\int_{0}^{5} (\cos x) \delta(x - 2\pi) \, dx\)
- The integral evaluates \(\cos x\) at \(x = 2\pi\), but since \(2\pi\) is outside the integration range [0, 5], the integral evaluates to zero.
Expert Solution
Step 1: Concept
I have use the concept property of dirac delta function
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