Final answer:
The antiderivative of 3e⁶⁶ is ½ e⁶⁶ + C, where C is the constant of integration. The basic integral rule for exponential functions was applied to find the antiderivative.
Step-by-step explanation:
The question asks for the antiderivative of 3e⁶⁶. An antiderivative, also known as an indefinite integral, of a function is another function whose derivative is the original function. In this case, we look for a function F(x) such that F'(x) equals 3e⁶⁶.
To find the antiderivative of 3e⁶⁶, we use the basic integral rule that the antiderivative of eˣ, where k is a constant, is ⅝ˣ plus a constant of integration, often denoted as C. Applying this rule:
∫ 3e⁶⁶ dx = ⅝⁶ e⁶⁶ + C = ½ e⁶⁶ + C
This is because when we differentiate ½ e⁶⁶ with respect to x, we get 6*½ e⁶⁶, which simplifies to 3e⁶⁶, thus confirming that ½ e⁶⁶ is indeed the antiderivative of 3e⁶⁶.