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

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

• A. e^x + C
• B. ln(e)x + C
• C. xe^x + C
• D. e^(2x) + C

Answer: (A)

The integral of \( e^x \) with respect to \( x \) can be determined by considering the unique property of the exponential function \( e^x \). When integrating \( e^x \), the result remains \( e^x \) itself due to the fact that the derivative of \( e^x \) is also \( e^x \). When performing the integral, we add a constant of integration \( C \) to account for any constant value that could have been present in the original function before differentiation. Hence, the integral \( \int e^x \, dx \) is expressed as: \[ \int e^x \, dx = e^x + C \] This accurately reflects the nature of exponential functions, providing a straightforward result without additional transformations or adjustments. The other choices imply different forms or operations that do not accurately represent the integral of \( e^x \). For example, \( \ln(e)x + C \) simplifies to \( x + C \), which is not the correct result of the integration. Similarly, \( xe^x + C \) and \( e^{2x} + C \) represent different integrals, not applicable in this context. Thus,

The integral of ( e^x ) with respect to ( x ) can be determined by considering the unique property of the exponential function ( e^x ). When integrating ( e^x ), the result remains ( e^x ) itself due to the fact that the derivative of ( e^x ) is also ( e^x ).

When performing the integral, we add a constant of integration ( C ) to account for any constant value that could have been present in the original function before differentiation. Hence, the integral ( \int e^x , dx ) is expressed as:

[

\int e^x , dx = e^x + C

]

This accurately reflects the nature of exponential functions, providing a straightforward result without additional transformations or adjustments.

The other choices imply different forms or operations that do not accurately represent the integral of ( e^x ). For example, ( \ln(e)x + C ) simplifies to ( x + C ), which is not the correct result of the integration. Similarly, ( xe^x + C ) and ( e^{2x} + C ) represent different integrals, not applicable in this context.

Thus,