Question

What is `f(x) = int e^(2x)-e^x+x dx` if `f(4 ) = 2`?

Answer 1

`f(x)=e^(2x)/2-e^x+x^2/2+e^4-e^8/2-6`

Explanation:

Let `I=int(e^(2x)-e^x+x)dx=e^(2x)/2-e^x+x^2/2+C`, `C` is a constant of integration.

To determine `C`, we are given the cond. that `f(4)=2`.

Now, `f(x)=I=e^(2x)/2-e^x+x^2/2+C.` Hence,
`f(4)=2 rArr e^8/2-e^4+8+C=2 rArr C=e^4-e^8/2-6.`

Sub.ing this value of `C` in `I`, we get,

`f(x)=e^(2x)/2-e^x+x^2/2+e^4-e^8/2-6`