Question

What is `f(x) = int (x+3)^2-3x dx` if `f(-1)=6`?

Answer 1

`f(x) = x^3/3 +3/2x^2 +9x + 83/6`

Explanation:

`f(x) = int (x+3)^2 -3x dx` if `f(-1) = 0`.

`f(-1)` is given to find the constant of integration and we shall use it after the integration.

`int (x+3)^2 - 3x dx`

To simplify things we can expand `(x+3)^2`

`(x+3)^2 = x^2+6x+9`

`int (x+3)^2 - 3x dx`
`=int x^2 + 6x + 9 - 3x dx`
`=int x^2 +3x + 9 dx`
`= x^(2+1)/(2+1) + 3x^(1+1)/(1+1) + 9x + C`
`=x^3/3 - 3x^2/2 + 9x + C`

This is `f(x) = x^3/3 + 3x^2/2 + 9x + C`

We are given `f(-1) = 6` Let us use this

`f(-1) = (-1)^3/3 +3(-1)^2/2 + 9(-1) + C`
`6 = -1/3+3/2 - 9 + C`
`6 = -2/6+9/6-54/6 + C`
`6=-47/6+C`
`6+47/6=C`
`36/6 + 47/6 = C`
`83/6 = C`

`f(x) = x^3/3 +3/2x^2 +9x + 83/6`