What is the integral of e^x?

The integral of ( e^x ) with respect to ( x ) is indeed expressed as ( e^x + C ), where ( C ) represents the constant of integration. This result arises from the fundamental properties of exponential functions. The unique characteristic of the exponential function ( e^x ) is that it is its own derivative, which means that when we take the integral—essentially working backward from differentiation—the process yields ( e^x ) again.

Adding the constant ( C ) is essential because integrals represent a family of functions that differ by a constant; thus, you need that constant term to encompass all possible antiderivatives of the function.

Other options do not reflect the correct integral of ( e^x ). For instance, the option involving subtraction of ( C ) does not adhere to the conventions of integration. The term ( xe^x ) is the form of a product rule application from differentiation, not applicable here for integration. Lastly, the logarithmic term ( \ln(e^x) ) simplifies to ( x ), which does not capture the essence of the integral we're calculating. Overall, ( e^x + C ) accurately denotes the integral of (