Final answer:
To find the value of x where F(x)=3 for the antiderivative F(x) = ∫_0^x e^{2t}dt, we set up and solve the equation (e^{2x} - 1)/2 = 3, resulting in x being equal to ln(7)/2.
Step-by-step explanation:
The student is asked to find the value of x such that the antiderivative F(x) = ∫_0^x e^{2t}dt equals 3. To solve for x, we differentiate F(x) by applying the Fundamental Theorem of Calculus, which tells us that F'(x) = e^{2x}. We then integrate e^{2x} from 0 to x and set the equation equal to 3:
F(x) = ∫_0^x e^{2t}dt = 3
Since ∫_0^x e^{2t}dt = (e^{2x} - 1)/2, we have:
(e^{2x} - 1)/2 = 3
Solving this equation for x, we get:
- e^{2x} - 1 = 6
- e^{2x} = 7
- 2x = ln(7)
- x = ln(7)/2
Thus, the value of x that makes F(x) equal to 3 is ln(7)/2.