Given that the antiderivative of f(x) = e4x is F(x) = 1/4e^4x+C, evaluate the integral from 0 to 2 of e^4x, dx.

e^8
1/4(1-e^8)
1/4(e^8-1)
1/4e^8 (MY ANNSWER)

∫[0,2] e^(4x) dx

= 1/4 e^(4x) + C [0,2]
= (1/4 e^8 + C)-(1/4 e^0 + C)
= 1/4 (e^8 - 1)

remember that e^0=1, not 0.

Well, well, well, look who's here! It's time for some integral fun, my friend! Now, let's get cracking and evaluate this integral.

Since the antiderivative of e^4x is 1/4e^4x + C, we can go ahead and compute the definite integral from 0 to 2 by subtracting the antiderivative evaluated at the upper limit (2) from the antiderivative evaluated at the lower limit (0).

So, plugging in the limits, we have:
F(2) - F(0) = 1/4e^(4*2) - 1/4e^(4*0)

Simplifying that, we get:
1/4e^8 - 1/4

Now, don't you worry, my friend, I won't leave it like that. Time to show off some more mathematical magic! Let's put our fractions together:

1/4e^8 - 1/4 = (1 - e^8)/4

So, the correct answer is 1/4(1 - e^8). Keep in mind, though, that you had a tiny typo in your answer. But hey, no worries! I'm here to point it out and keep the smiles going. Keep up the good work, my friend!

To evaluate the integral from 0 to 2 of e^4x, dx using the given antiderivative F(x) = 1/4e^4x+C, we can apply the second part of the Fundamental Theorem of Calculus.

Using the antiderivative, F(x) = 1/4e^4x+C, we can find the definite integral from a to b using the formula:

∫[a to b] f(x) dx = F(b) - F(a)

Plugging in the values for a and b, which are 0 and 2 respectively, we have:

∫[0 to 2] e^4x dx = F(2) - F(0)

Now let's plug in the values into F(x) = 1/4e^4x+C:

F(2) = 1/4e^4(2)+C
F(2) = 1/4e^8+C

F(0) = 1/4e^4(0)+C
F(0) = 1/4+C

Substituting these values back into the definite integral formula:

∫[0 to 2] e^4x dx = F(2) - F(0)
∫[0 to 2] e^4x dx = (1/4e^8) - (1/4)

Simplifying further, we get:

∫[0 to 2] e^4x dx = 1/4(e^8-1)

So, the correct answer is 1/4(e^8-1).

To evaluate the integral from 0 to 2 of e^4x, dx, you can use the fundamental theorem of calculus. According to the theorem, the definite integral of a function over an interval can be found by subtracting the antiderivative at the lower limit from the antiderivative at the upper limit.

In this case, the antiderivative of f(x) = e^4x is given as F(x) = 1/4e^4x + C, where C represents the constant of integration.

To evaluate the integral, substitute the upper limit (2) and the lower limit (0) into the antiderivative expression and subtract the results:

F(2) - F(0) = (1/4e^4(2)) - (1/4e^4(0))
= (1/4e^8) - (1/4)
= 1/4e^8 - 1/4
= 1/4(e^8 - 1)

So, the correct answer is 1/4(e^8 - 1).