For the integral ∫ e^(kx) dx, where k is a constant, what is the result?

For the integral ∫ e^(kx) dx, where k is a constant, what is the result?

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

Answer: (A)

To find the result of the integral ∫ e^(kx) dx, where k is a constant, we use a standard technique for integrating exponential functions. When integrating e^(kx), we apply the rule that states when integrating an exponential function of the form e^(ax) with respect to x, the integral can be expressed as (1/a)e^(ax) + C, where a is a constant, and C is the constant of integration. In this case, a is replaced by k. Therefore, we follow the rule: ∫ e^(kx) dx = (1/k)e^(kx) + C. This properly accounts for the factor k in the exponent. The division by k is necessary to ensure that the differentiation of the result returns us to the original function. If we take the derivative of (1/k)e^(kx) with respect to x, we get e^(kx) back, confirming that the integral works as intended. Thus, (1/k)e^(kx) + C is indeed the correct form of the integral for ∫ e^(kx) dx.

To find the result of the integral ∫ e^(kx) dx, where k is a constant, we use a standard technique for integrating exponential functions.

When integrating e^(kx), we apply the rule that states when integrating an exponential function of the form e^(ax) with respect to x, the integral can be expressed as (1/a)e^(ax) + C, where a is a constant, and C is the constant of integration.

In this case, a is replaced by k. Therefore, we follow the rule:

∫ e^(kx) dx = (1/k)e^(kx) + C.

This properly accounts for the factor k in the exponent. The division by k is necessary to ensure that the differentiation of the result returns us to the original function. If we take the derivative of (1/k)e^(kx) with respect to x, we get e^(kx) back, confirming that the integral works as intended.

Thus, (1/k)e^(kx) + C is indeed the correct form of the integral for ∫ e^(kx) dx.