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

1 Answer | Add Yours

sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

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` .

We’ve answered 318,986 questions. We can answer yours, too.

Ask a question