Calculus question about antiderivatives

I understand the process of antiderivatives, but I got stuck on this problem...please help!!

So for this problem it says: find the antiderivative F(x) with F'(x) = f(x) and F(0)=5.

6e^3x

I'm not even sure where to begin with this problem?? Please help! Thank you!:)

Hey! :)

So your original function is f(x)=6e^3x, and you've been given the initial condition F(0)=5. Therefore, this is an initial value problem. The first thing you want to do is find the antiderivative of f(x).

Integral of 6e^3x dx

Bring the 6 out in front of the integral

6 (integral of e^3x dx)

The integral of e^3x is e^3x, but Chain Rule tells you that you have to divide by the derivative of the exponent. The exponent is 3x, and it's derivative is 3, so you have to divide by 3. Therefore, the integral is

(6/3)e^3x+C, which simplifies to

F(x)=2e^3x+C

Now your initial condition tells you to plug in 0 for x and 5 for F(x). The goal here is to solve for C, the constant of integration.

5=2e^3(0)+C
5=2(1)+C (anything raised to the 0 power is 1)
5=2+C
C=3

Plugging C back into your antiderivative gives you a final answer of

F(x)=2e^3x+3

I really hope that helps, and good luck to you! :)

it wants the original function, f(x), by finding the value of c

∫(6e^3x)dx = 2e^3x + c

F(0)=5; it means x of the integral is 0 & y is 5

5 = 2e^3(0) + c

5 = 2 + c; c = 3

so, f(x) = 2e^3x + 3