What is the integral of e^(2x) with respect to x?

What is the integral of e^(2x) with respect to x?

• A. (1/2)e^(2x) + C
• B. e^(2x) + C
• C. (1/3)e^(2x) + C
• D. (1/4)e^(2x) + C

Answer: (A)

To find the integral of \( e^{2x} \) with respect to \( x \), we can apply the technique for integrating exponential functions. The general form for the integral of \( e^{kx} \) is given by: \[ \int e^{kx} \, dx = \frac{1}{k} e^{kx} + C \] where \( k \) is a constant, and \( C \) is the integration constant. In this case, \( k \) is 2 since we have \( e^{2x} \). Therefore, when applying the formula, we substitute \( k \) with 2: \[ \int e^{2x} \, dx = \frac{1}{2} e^{2x} + C \] This shows that the correct integral result is indeed \( (1/2)e^{2x} + C \). The constant factor \( \frac{1}{2} \) arises from dividing by \( k \), which in this case is 2, making it necessary to appropriately scale the exponential function after integration to ensure that the derivative of the integrated function yields the original function \( e^{2x} \).

To find the integral of ( e^{2x} ) with respect to ( x ), we can apply the technique for integrating exponential functions. The general form for the integral of ( e^{kx} ) is given by:

[

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

]

where ( k ) is a constant, and ( C ) is the integration constant.

In this case, ( k ) is 2 since we have ( e^{2x} ). Therefore, when applying the formula, we substitute ( k ) with 2:

[

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

]

This shows that the correct integral result is indeed ( (1/2)e^{2x} + C ).

The constant factor ( \frac{1}{2} ) arises from dividing by ( k ), which in this case is 2, making it necessary to appropriately scale the exponential function after integration to ensure that the derivative of the integrated function yields the original function ( e^{2x} ).