Answer:
This is the same as writing -0.5(x^2+1)^5 + 2.5
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Explanation:
We take the integral to find the original f(x) function. Another term for "integral" is "antiderivative", which basically is the inverse of a derivative. There's more to the story, but this is the basic idea.
Apply a u-substitution
u = x^2+1
du/dx = 2x
du = 2xdx
xdx = du/2 = 0.5du
So,
This means that
To determine the constant C, we plug in x = 0 and replace f(x) with 2.
This is directly from the fact that f(0) = 2.
Therefore,
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Checking the answer:
Plug in x = 0. You should find that f(0) = 2. I'll let you do these steps.
Apply the derivative and use the chain rule to find:
The answer is fully confirmed. Another way you can confirm the answer is to use free tools like GeoGebra or WolframAlpha. I show an example of this in the screenshot below.