Given F antiderivative of f = 4x+3/x, what is F(-e) if F(-1)=0?

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Using the definition of anti-derivative of function `f(x)` yields:

`F(x) + c = int f(x)dx`

By this definition, you need to evaluate the indefinite integral of the function `f(x) = 4x + 3/x` , such that:

`F(x) + c = int (4x + 3/x) dx`

Using the property of linearity of indefinite integral, yields:

`F(x) + c = int (4x) dx + int (3/x) dx`

`F(x) + c = 4x^2/2 + 3ln|x|`

`F(x) + c = 2x^2 + 3ln|x| => 2x^2 + 3ln|x| + c = F(x)`

The problem provides the information that `F(-1) = 0` , hence, you need to replace -1 for x in equation of `F(x)` , such that:

`F(-1) = 2(-1)^2 + 3ln|-1| + c => F(-1) = 2 + 3ln 1 + c`

Since `ln 1 = 0` yields:

`F(-1) = 2 + c => 0 = 2 + c => c = -2`

Once the evaluation of the constant c is completed, you may evaluate `F(-e)` such that:

`F(-e) = 2(-e)^2 + 3ln|-e| - 2`

`F(-e) = 2e^2 + 3ln e - 2`

Since ` ln e = 1` yields:

`F(-e) = 2e^2 + 3 - 2 => F(-e) = 2e^2 + 1`

**Hence, evaluating the anti-derivative `F(x)` , at `x = -e` , under the given conditions, yields `F(-e) = 2e^2 + 1` .**

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