What is the integral of e^(4x) dx?

What is the integral of e^(4x) dx?

• A. (1/4)e^(4x) + C
• B. (1/3)e^(4x) + C
• C. (1/5)e^(4x) + C
• D. e^(4x) + C

Answer: (A)

To find the integral of e^(4x) with respect to x, we can use the basic rule for integrating exponential functions. The integral of e^(kx), where k is a constant, is given by (1/k)e^(kx) + C, where C represents the constant of integration. In the case of e^(4x), the constant k is equal to 4. Thus, applying the integral formula: \[ \int e^{4x} \, dx = \frac{1}{4} e^{4x} + C \] This demonstrates that the correct integral is indeed (1/4)e^(4x) + C. This option correctly reflects the application of the integration rule for exponential functions where we divide by the constant multiplying x in the exponent, which reinforces the integral's dependency on the value of k. Therefore, the choice that states (1/4)e^(4x) + C accurately represents the integral of e^(4x).

To find the integral of e^(4x) with respect to x, we can use the basic rule for integrating exponential functions. The integral of e^(kx), where k is a constant, is given by (1/k)e^(kx) + C, where C represents the constant of integration.

In the case of e^(4x), the constant k is equal to 4. Thus, applying the integral formula:

[

\int e^{4x} , dx = \frac{1}{4} e^{4x} + C

]

This demonstrates that the correct integral is indeed (1/4)e^(4x) + C.

This option correctly reflects the application of the integration rule for exponential functions where we divide by the constant multiplying x in the exponent, which reinforces the integral's dependency on the value of k. Therefore, the choice that states (1/4)e^(4x) + C accurately represents the integral of e^(4x).