What is the integral of e^(2x) with respect to x?

To find the integral of ( e^{2x} ) with respect to ( x ), we can apply the technique for integrating exponential functions. The general form for the integral of ( e^{kx} ) is given by:

[

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

]

where ( k ) is a constant, and ( C ) is the integration constant.

In this case, ( k ) is 2 since we have ( e^{2x} ). Therefore, when applying the formula, we substitute ( k ) with 2:

[

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

]

This shows that the correct integral result is indeed ( (1/2)e^{2x} + C ).

The constant factor ( \frac{1}{2} ) arises from dividing by ( k ), which in this case is 2, making it necessary to appropriately scale the exponential function after integration to ensure that the derivative of the integrated function yields the original function ( e^{2x} ).