What is the integral of e^(2x) dx?

What is the integral of e^(2x) dx?

• A. (1/2)e^(2x) + C
• B. (1/3)e^(3x) + C
• C. e^(2x) + C
• D. 2e^(2x) + C

Answer: (A)

To find the integral of \( e^{2x} \, dx \), we can utilize the basic integration rule for exponential functions. The general rule states that the integral of \( e^{kx} \) with respect to \( x \) is given by \( \frac{1}{k} e^{kx} + C \), where \( k \) is a constant and \( C \) represents the constant of integration. In this case, we have \( k = 2 \). Following the rule, we compute the integral as follows: \[ \int e^{2x} \, dx = \frac{1}{2} e^{2x} + C \] This result indicates that the factor of \( \frac{1}{2} \) in front of \( e^{2x} \) is essential because the derivative of \( \frac{1}{2} e^{2x} \) will yield \( e^{2x} \) when using the chain rule. Consequently, the integral accumulates into the final correct answer, which is \( \frac{1}{2} e^{2x} + C \). The other options do not represent the correct application of the integration rule

To find the integral of ( e^{2x} , dx ), we can utilize the basic integration rule for exponential functions. The general rule states that the integral of ( e^{kx} ) with respect to ( x ) is given by ( \frac{1}{k} e^{kx} + C ), where ( k ) is a constant and ( C ) represents the constant of integration.

In this case, we have ( k = 2 ). Following the rule, we compute the integral as follows:

[

\int e^{2x} , dx = \frac{1}{2} e^{2x} + C

]

This result indicates that the factor of ( \frac{1}{2} ) in front of ( e^{2x} ) is essential because the derivative of ( \frac{1}{2} e^{2x} ) will yield ( e^{2x} ) when using the chain rule. Consequently, the integral accumulates into the final correct answer, which is ( \frac{1}{2} e^{2x} + C ).

The other options do not represent the correct application of the integration rule