Compute the integral ∫ e^(3x) dx.

Compute the integral ∫ e^(3x) dx.

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

Answer: (A)

To compute the integral ∫ e^(3x) dx, we can use the formula for integrating exponential functions, which states that ∫ e^(kx) dx = (1/k)e^(kx) + C, where k is a constant and C is the constant of integration. In this case, we have k = 3. Applying the formula: ∫ e^(3x) dx = (1/3)e^(3x) + C This result shows that we multiply the exponential function e^(3x) by the reciprocal of the constant in the exponent, which in this case is 1/3, followed by the integration constant C. This correctly identifies the behavior of the exponential function when integrated with respect to x. Thus, the answer reflects the appropriate manipulation for an exponential function where the coefficient of x in the exponent affects the result of the integral directly.

To compute the integral ∫ e^(3x) dx, we can use the formula for integrating exponential functions, which states that ∫ e^(kx) dx = (1/k)e^(kx) + C, where k is a constant and C is the constant of integration.

In this case, we have k = 3. Applying the formula:

∫ e^(3x) dx = (1/3)e^(3x) + C

This result shows that we multiply the exponential function e^(3x) by the reciprocal of the constant in the exponent, which in this case is 1/3, followed by the integration constant C.

This correctly identifies the behavior of the exponential function when integrated with respect to x. Thus, the answer reflects the appropriate manipulation for an exponential function where the coefficient of x in the exponent affects the result of the integral directly.