How do you integrate e^(kx) where k is a constant?

Integrating the expression ( e^{kx} ), where ( k ) is a constant, relies on understanding how the exponential function behaves under integration.

When you perform the integration of ( e^{kx} ), you apply a basic rule of integration for exponential functions. The standard result for integrating ( e^{ax} ) (where ( a ) is any constant) is ( \frac{1}{a} e^{ax} + C ). In this case, ( a ) is represented by ( k ). Thus, integrating ( e^{kx} ) gives:

[

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

]

Here, ( \frac{1}{k} ) is a scaling factor that arises because of the derivative of the exponent ( kx ). When this function is differentiated, ( k ) is a factor that appears, and hence when you integrate, you need to counteract that by dividing by ( k ).

Therefore, the correct integration result, which includes the constant of integration ( C ), is ( \frac{1}{k} e^{kx} + C ). This