Question

What is the value of the integral ∫(x / eˣ-)) dx from 0 to infinity?

a) 0
b) 1
c) -1
d) [infinity]

Answer 1

The value of the integral ∫(x / eˣ-) dx from 0 to infinity is 0.

Step-by-step explanation:

The value of the integral ∫(x / eˣ-) dx from 0 to infinity is 0.

To evaluate this integral, we can use integration by parts. Let's assign u = x and dv = eˣ- dx. Taking the derivative of u, we get du = dx, and integrating dv, we get v = -eˣ. Now we can use the formula for integration by parts: ∫u dv = uv - ∫v du. Applying this formula to our integral, we have:

∫(x / eˣ-) dx = -xeˣ + ∫eˣ dx.

This simplifies to -xeˣ + eˣ + C, where C is the constant of integration. Evaluating this expression from 0 to infinity, we get:

[∫(x / eˣ-) dx] from 0 to infinity = (-∞e^(-∞) + e^(-∞) + C) - (0e^0 + e^0 + C) = (0 + e^(-∞) + C) - (0 + 1 + C) = 0.