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

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

• A. -e^(-x) + C
• B. e^(-x) + C
• C. (1/2)e^(-x) + C
• D. (-1)e^(-x) + C

Answer: (A)

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

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