Question

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

Answer 1

`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.