Final answer:
The area under the curve of f(x) = 6x + eˣ on [1, ln(8)] is 35 - e.
Step-by-step explanation:
To find the area under the curve for the function f(x) = 6x + eˣ on the interval [1, ln(8)], we can use integration. The integral of the function represents the area under the curve between the given bounds.
First, we need to find the antiderivative of f(x), which is F(x) = 3x² + eˣ. Then, we evaluate the definite integral of F(x) from 1 to ln(8). This can be written as:
∫[1, ln(8)] (6x + eˣ) dx = [3x² + eˣ] from 1 to ln(8)
Plugging in the upper and lower bounds, the area under the curve is:
[(3(ln(8))² + e^(ln(8)))] - [(3(1)² + e^(1))] = [24 + 8] - [3 + e] = 35 - e