Find the integral of e^(2x) dx.

To solve the integral of e^(2x) with respect to x, we use the method of substitution or direct integration recognition.

The integral can be approached by recognizing that the integrand, e^(2x), is an exponential function. When integrating an exponential function of the form e^(kx), the general formula is:

∫ 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 = 2. Thus, applying the formula:

∫ e^(2x) dx = (1/2)e^(2x) + C.

This shows that the factor in front of e^(2x) corresponds to the reciprocal of the coefficient of x in the exponent. Therefore, the correct response is indeed (1/2)e^(2x) + C, which reflects this relationship accurately.

The other options provided do not take the coefficient of x into account correctly:

• e^(2x) + C implies a multiplicative factor of 1 instead of the required 1/2.

• 2e^(2x) + C suggests an erroneous multiplication by 2, ignoring the integration rule.

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