Which of the following represents the integral of e^(-x) dx?

To find the integral of ( e^{-x} , dx ), we can utilize the property of integration that involves exponential functions. The integral of ( e^{kx} ) with respect to ( x ) is given by the formula:

[

\int e^{kx} , dx = \frac{1}{k} e^{kx} + C

]

In this case, we have ( k = -1 ). Applying this formula, we can calculate the integral:

[

\int e^{-x} , dx = \frac{1}{-1} e^{-x} + C = -e^{-x} + C

]

This means that the integral of ( e^{-x} ) is indeed ( -e^{-x} + C ), where ( C ) is the constant of integration.

Thus, the correct representation of the integral is ( -e^{-x} + C ). The inclusion of the constant ( C ) is critical, as it represents the family of functions that differ by a constant, which is standard in indefinite integrals. This aligns perfectly with option A, confirming it as the right choice.

The other options do not accurately reflect the integration