What is f(x) = int e^(2x-1)-e^(1-2x)+e^x dx if f(2) = 3 ?

1 Answer
Feb 20, 2017

The function f is:

f (x) = 1/2 (e^{2x-1} + e^{1-2x}) + e^x + 3 - e^2 - 1/2 (e^3 + e^{- 3}).

Explanation:

By integrating the value of f:

f (x) = int (e^{2 x - 1} - e^{1 - 2x} + e^x) dx =

= int e^{2x-1} dx - int e^{1-2x} dx + int e^x dx =

= 1/2 e^{2x-1} - (- 1/2) e^{1 - 2x} + e^x + C =

= 1/2 (e^{2x-1} + e^{1-2x}) + e^x + C

Then, substituting the known point of f:

f (2) = 1/2 (e^3 + e^{- 3}) + e^2 + C = 3,

we can obtain the value of the constant C:

C = 3 - e^2 - 1/2 (e^3 + e^{- 3}).

The function f will have the following formula:

f (x) = 1/2 (e^{2x-1} + e^{1-2x}) + e^x + 3 - e^2 - 1/2 (e^3 + e^{- 3}).