Question

Evaluate the integral ∫[infinity] π/2 dxδ(sin x)e−x. What is the result?

a) 0
b) π/2
c) -1
d) e^(-π/2)

Answer 1

To evaluate the integral ∫[infinity] π/2 dxδ(sin x)e−x, we can integrate by parts. The result is 0.

Step-by-step explanation:

To evaluate the integral ∫[infinity] π/2 dxδ(sin x)e−x, we can integrate by parts. Let's define u = sin(x) and dv = e^(-x)dx. Taking the derivatives and integrals, we have du = cos(x)dx and v = -e^(-x).

Using the formula for integration by parts, ∫u dv = uv - ∫v du, we can rewrite the integral as:

∫[infinity] π/2 dxδ(sin x)e−x = -sin(x)e^(-x)∣[infinity] π/2 - ∫[infinity] π/2 -e^(-x)cos(x) dx.

Next, we can simplify the second part by integrating by parts again with u = cos(x) and dv = -e^(-x)dx. After finding the derivatives and integrals, we have du = -sin(x)dx and v = e^(-x).

Using the integration by parts formula, the second part becomes:

-e^(-x)cos(x)∣[infinity] π/2 + ∫[infinity] π/2 e^(-x)sin(x) dx.

We can see that the two parts of the integral cancel each other out when evaluated from π/2 to infinity. Therefore, the result is 0. Hence, the answer is a) 0.