Question

What is `f(x) = int x^3-x` if `f(2)=4`?

Answer 1

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

Explanation:

For solving the problem first integrate the indefinite integral.

`f(x)=int quad x^3-x dx`

Use rule `int quad x^n dx = x^(n+1)/(n+1) + C`

`f(x)=x^4/4 - x^2/2 + C`

We are given `f(2)=4`
We are going to find `C` using the given value.

`f(2)=(2)^4/4-(2)^2/2+C`
`4=16/4-4/2+C`
`4=4-2+C`
#4=2+C #C=2#

Therefore,

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

Answer 2

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

Explanation:

Given: `f(x) = int(x^3-x)` `dx` and `f(2)=4`

`f(x)=int(x^3-x)` `dx` = `x^4/4-x^2/2+C`

`f(x)=x^4/4-x^2/2+C`

`f(2)=4=2^4/4-2^2/2+C`

and `C=2`

therefore `f(x)=x^4/4-x^2/2+2`