Question

Evaluate the integral ∫ 0^3 12e^(3x) dx

Answer 1

To evaluate the integral of 12e^(3x) from 0 to 3, we find the antiderivative, apply the limits, and subtract to get the final answer, which is 4e^(9) - 4. F(x) is called an antiderivative or Newton-Leibnitz integral or primitive of a function f(x) on an interval I. F'(x) = f(x), for every value of x in I. Integral is the representation of the area of a region under a curve. We approximate the actual value of an integral by drawing rectangles.

Step-by-step explanation:

The question asks to evaluate the integral of the function 12e^(3x) with respect to x from 0 to 3. To tackle this problem, we use the fact that the integral of e^(ax) with respect to x is (1/a)*e^(ax), plus a constant of integration when dealing with an indefinite integral.

1. Integrate the function 12e^(3x) with respect to x which gives us (12/3)*e^(3x) since 3 is the coefficient of x in the exponent.
2. The simplified form is 4e^(3x), and we are looking for the definite integral from the lower limit of 0 to the upper limit of 3.
3. Applying the Fundamental Theorem of Calculus, we evaluate 4e^(3x) at the upper limit of x = 3 and subtract the evaluation at the lower limit of x = 0.
4. The final answer is obtained by plugging in these limits into the integral expression. Evaluating at the upper limit gives us 4e^(9), and at the lower limit gives us 4e^(0), which is just 4.
5. Therefore, the definite integral is 4e^(9) - 4.