What is the integral of e^(kx) dx, where k is a constant?

What is the integral of e^(kx) dx, where k is a constant?

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

Answer: (A)

The integral of \( e^{kx} \, dx \) can be derived from the rule of integrating exponential functions. When integrating \( e^{kx} \), where \( k \) is a constant, we can apply the technique of substitution or use knowledge of the form of exponential integrals directly. The derivative of \( e^{kx} \) with respect to \( x \) is \( k e^{kx} \). This indicates that when we integrate \( e^{kx} \), we have to account for the factor \( k \) that arises from the differentiation. Therefore, to reverse this process, we effectively divide by \( k \) in the integral. Thus, the integral can be formulated as: \[ \int e^{kx} \, dx = \frac{1}{k} e^{kx} + C \] where \( C \) is the constant of integration. This confirms that the correct choice states \( \frac{1}{k} e^{kx} + C \) as the result of the integral. Understanding this integration rule is essential because it extends to further applications in calculus, where integrals of exponential functions frequently arise. Recognizing that the presence of a constant multiplier in

The integral of ( e^{kx} , dx ) can be derived from the rule of integrating exponential functions. When integrating ( e^{kx} ), where ( k ) is a constant, we can apply the technique of substitution or use knowledge of the form of exponential integrals directly.

The derivative of ( e^{kx} ) with respect to ( x ) is ( k e^{kx} ). This indicates that when we integrate ( e^{kx} ), we have to account for the factor ( k ) that arises from the differentiation. Therefore, to reverse this process, we effectively divide by ( k ) in the integral.

Thus, the integral can be formulated as:

[

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

]

where ( C ) is the constant of integration. This confirms that the correct choice states ( \frac{1}{k} e^{kx} + C ) as the result of the integral.

Understanding this integration rule is essential because it extends to further applications in calculus, where integrals of exponential functions frequently arise. Recognizing that the presence of a constant multiplier in