Question

Evaluate the integral:
\[ \int_{0}^{\infty} e^{-x} \delta(\lambda^2 - 4)dx \]

• A. \( \frac{1}{4e^2} \)
• B. \( 1 \)
• C. \( \frac{1}{4e^2} \)
• D. \( \frac{1}{2} (e^{-2} + e^2) \)

Hint:

Use the Dirac delta function property for evaluating integrals.

Answer 1

The Correct Option is A

Using the sifting property of the Dirac delta function:
\[ \int f(x) \delta(g(x)) dx = \sum_i \frac{f(x_i)}{|g'(x_i)|} \] where \( g(\lambda) = \lambda^2 - 4 \) gives \( \lambda = \pm2 \). Evaluating for \( \lambda = 2 \), we get:
\[ \frac{e^{-2}}{2 \cdot 2} = \frac{1}{4e^2} \]