What is the result of ∫ e^(kx) dx where k is a constant?

What is the result of ∫ e^(kx) dx where k is a constant?

• A. (1/k)e^(kx) + C
• B. ke^(kx) + C
• C. e^(kx) + C
• D. (1/k^2)e^(kx) + C

Answer: (A)

The integral ∫ e^(kx) dx represents the process of finding the antiderivative of the exponential function e^(kx), where k is a constant. The basic rule for integrating exponential functions states that the integral of e^(ax) with respect to x is (1/a)e^(ax) + C, where C is the constant of integration. In this case, the variable 'a' corresponds to 'k'. Applying this rule, we have: ∫ e^(kx) dx = (1/k)e^(kx) + C. This result makes sense because when differentiating (1/k)e^(kx), the constant '1/k' remains, and by applying the chain rule, the derivative gives back e^(kx). Therefore, the factor of (1/k) correctly adjusts for the rate of change introduced by the k constant in the exponent. This reasoning confirms that the expression (1/k)e^(kx) + C is the correct antiderivative of e^(kx) when k is a constant. Consequently, the selected answer is valid according to integration rules for exponential functions.

The integral ∫ e^(kx) dx represents the process of finding the antiderivative of the exponential function e^(kx), where k is a constant. The basic rule for integrating exponential functions states that the integral of e^(ax) with respect to x is (1/a)e^(ax) + C, where C is the constant of integration.

In this case, the variable 'a' corresponds to 'k'. Applying this rule, we have:

∫ e^(kx) dx = (1/k)e^(kx) + C.

This result makes sense because when differentiating (1/k)e^(kx), the constant '1/k' remains, and by applying the chain rule, the derivative gives back e^(kx). Therefore, the factor of (1/k) correctly adjusts for the rate of change introduced by the k constant in the exponent.

This reasoning confirms that the expression (1/k)e^(kx) + C is the correct antiderivative of e^(kx) when k is a constant. Consequently, the selected answer is valid according to integration rules for exponential functions.