# Given F primitive of real f(x) = -x+11)/(x^2-x-2), F(3)=-8ln2, what is F(8)? A. F(8) =2ln3-3ln5 B. F(8)=3ln2-5ln3 C. F(8)=2ln5-3ln2 D. F(8)=3ln5-5ln2

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You need to evaluate the primitive `F(x)` , hence, you need to evaluate the indefinite integral of `f(x)` , such that:

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

`F(x) = int (-x+11)/(x^2-x-2) dx`

Using the property of linearity of integral, yields:

`F(x) = int -x/(x^2-x-2) dx + 11 int 1//(x^2-x-2) dx`

You should evaluate the first integral ` int -x/(x^2-x-2) dx` using the following substitution, such that:

`x^2-x-2 = t => (2x - 1)dx = dt`

`-int x/(x^2-x-2) dx = -(1/2)int (2x - 1 + 1)/(x^2-x-2) dx`

Using the property of linearity of integral, yields:

`-int x/(x^2-x-2) dx = -(1/2)int (2x - 1)/(x^2-x-2) dx - (1/2)int 1/(x^2-x-2) dx`

`F(x) = -(1/2)int (2x - 1)/(x^2-x-2) dx - (1/2)int 1/(x^2-x-2) dx + 11 int 1//(x^2-x-2) dx`

`F(x) = -(1/2)ln|x^2 - x - 2| + (21/2)int 1/(x^2-x-2) dx`

You need to evaluate the integral `int 1//(x^2-x-2) dx` using the partial fraction decomposition, such that:

`1/(x^2-x-2) = 1/((x - 2)(x + 1))`

`1/((x - 2)(x + 1)) = a/(x - 2) + b/(x + 1 )`

`1 = ax + a + bx - b => 1 = x(a + b) + a - 2b`

Equating the coefficients of equal powers yields:

`{(a+b=0),(a-2b=1):} => a = 1/3, b = -1/3`

`1/((x - 2)(x + 1)) = 1/(3(x - 2)) - 1/(3(x + 1 ))`

Integrating both sides, yields:

`int 1/((x - 2)(x + 1)) dx = int 1/(3(x - 2)) dx - int 1/(3(x + 1 )) dx`

`int 1/((x - 2)(x + 1)) dx = 1/3(ln|x - 2| - ln|x+1|) + c`

Using the logarithmic identities, yields:

`int 1/((x - 2)(x + 1)) dx = 1/3(ln|(x-2)/(x+1)|) + c`

`F(x) = -(1/2)ln|x^2 - x - 2| + (21/6)(ln|(x-2)/(x+1)|) + c`

The problem provides the information that `F(3) = -8ln 2` , such that:

`F(3) = -(1/2)ln|3^2 - 3 - 2| + (21/6)(ln|(3-2)/(3+1)|) + c `

`F(3) = -(1/2)ln 4 + (21/6)ln(1/4) + c => F(3) = -(1/2)ln 4 + (21/6)ln 1 - (21/6)ln 4 + c`

Since `ln 1 = 0` yields:

`F(3) = -(1/2)ln 4 - (21/6)ln 4 => F(3) = -4ln 4`

Replacing -`8ln 2` for `F(3)` yields:

`-8ln 2 = - 4ln 4 + c => c = -8ln 2 + 4ln 4 => c = -8ln 2 + 8ln 2 = 0`

You need to evaluate `F(8)` , such that:

`F(8)= -(1/2)ln|8^2 - 8 - 2| + (21/6)(ln|(8-2)/(8+1)|)`

`F(8)= -(1/2)ln 54 + (21/6)(ln(6/9))`

`F(8)= -(1/2)ln (6*9) + (21/6)ln 6 - (21/6)ln 9`

`F(8)= -(1/2)ln6 - (1/2)ln9 + (21/6)ln 6 - (21/6)ln 9`

`F(8)= 3ln 6 - 4ln 9`

`F(8)= 3ln(2*3) - 4ln(3^2) => F(8)= 3ln2 + 3ln3 - 8ln3`

`F(8) = 3ln2 - 5ln3`

Hence, evaluating the primitive ` F(x)` at `x = 8` , yields `F(8)= 3ln2 - 5ln3` , hence, you need to select the answer B.