antiderivatives calculus

Question

Determine the antiderivative F that satisfies the following conditions.

(a) f(x) = 6x^3 − 5x + 2, F(−2) = 8

(b) f(x) = 2/x − e^3x + √ x, F(1) = 1

Answer 1

When it's telling you to determine the antiderivative in this way, it's telling you to solve for a constant of integration.

So for example, in your first problem.

f(x)=6x^3 − 5x + 2. So F(x) = 6x⁴/4 - 5x²/2 +2x + C.

using our condition that F(-2)=8, we may say that:

8 = 6*16/4 - 20/2 -4 +C.

= 24 - 10 - 4 + C

= 10+C.

Rearranging for C, we get C = -2.

So your final answer is F(x) = 6x⁴/4 - 5x²/2 +2x -2.

NOTE: it's only a coincidence that our constant of integration turned out to be equivalent to our input point of -2.