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

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

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

Answer: (A)

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

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