What is the integral of e^(2x) dx?

To find the integral of ( e^{2x} , dx ), we can utilize the basic integration rule for exponential functions. The general rule states that the integral of ( e^{kx} ) with respect to ( x ) is given by ( \frac{1}{k} e^{kx} + C ), where ( k ) is a constant and ( C ) represents the constant of integration.

In this case, we have ( k = 2 ). Following the rule, we compute the integral as follows:

[

\int e^{2x} , dx = \frac{1}{2} e^{2x} + C

]

This result indicates that the factor of ( \frac{1}{2} ) in front of ( e^{2x} ) is essential because the derivative of ( \frac{1}{2} e^{2x} ) will yield ( e^{2x} ) when using the chain rule. Consequently, the integral accumulates into the final correct answer, which is ( \frac{1}{2} e^{2x} + C ).

The other options do not represent the correct application of the integration rule