Question

What is `f(x) = int (x-2)^2-5x+4 dx` if `f(2) = 1`?

Answer 1

`f(x) = 1/3 (x-2)^3 -5/2 x^2 + 4x + 3`

Explanation:

`intax^n dx =(ax^(n+1))/(n+1) + c`

is the standard integral , where c is the constant of integration

further `int(ax+b)^n dx = (ax+b)^(n+1)/(a(n+1)) + c`

These can be applied 'term by term' to the above integral

`f(x) = 1/3 (x-2)^3 - 5/2 x^2 + 4x + c`

using f(2) = 1 , allows c , to be found

`1/3 (2-2)^3 - 5/2. 4 + 8 + c = 1`

0 - 10 + 8 + c = 1 → c = 3

`rArr f(x) = 1/3 (x-2)^3 - 5/2 x^2 + 4x + 3`