# Find a function f(x) with the following property: f(x) = 2e^2x - 4/x + 3/x^2, f(1) = -2calculus

sciencesolve | Teacher | (Level 3) Educator Emeritus

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Supposing that you want to find the function F(x) that has as derivative the function f(x), you need to integrate the function f(x) such that:

`int f(x) dx = int (2e^2x - 4/x + 3/x^2) dx`

You need to split the integral in three simpler integrals such that:

`int (2e^2x - 4/x + 3/x^2) dx = int 2e^2x dx- int 4/x dx+ int 3/x^2 dx`

`int (2e^2x - 4/x + 3/x^2) dx = 2*e^(2x)/2 - 4ln|x| + 3 x^(-2+1)/(-2+1) + c`

`int (2e^2x - 4/x + 3/x^2) dx = e^(2x) - 4ln|x|- 3/x + c`

The problem provides the information that F(1)=-2, hence substituting 1 for x in equation  `e^(2x) - 4ln|x| - 3/x + c = F(x)`  you may find the constant term such that:

`F(1) = e^2 - 4*ln 1 - 3/1 + c`

`-2 = e^2 - 4*0 - 3 + c =gt c = 3 - 2 - e^2`

`c = 1 - e^2`

Hence, evaluating the function F(x) under given conditions yields `F(x) = e^(2x) - 4ln|x| - 3/x + 1 - e^2` .