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

To find the integral of ( e^{-2x} ) with respect to ( x ), we can utilize the rule for integrating exponential functions.

When integrating ( e^{kx} ), where ( k ) is a constant, the integral can be computed using the formula:

[

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

]

In this case, the integral we are dealing with is ( e^{-2x} ). Here, ( k = -2 ). Applying the formula, we have:

[

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

]

This matches the first choice, highlighting that the factor in front of the exponential function is (-\frac{1}{2}) due to the negative coefficient from the exponent. The constant (C) is included because when performing an indefinite integral, we need to account for all antiderivatives.

The reasoning confirms that the integral indeed equals ((-1/2)e^{-2x} + C\