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

1 Answer

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

Explanation:

First we have to integrate

int(e^(2x-1)-e^(3-x)+e^(-x))dx

the integral is

f(x)=1/2e^(2x-1)+e^(3-x)-e^(-x)+C

but f(2)=3

so, it follows

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

Solving for C integration constant

C=-e^3/2-e+e^(-2)+3

Our final answer

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

God bless....I hope the explanation is useful.