Question

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

Answer 1

`7/3`

Explanation:

`int(x-3)^2-3x+4 dx`
`= (x-3)^3/3-(3x^2)/2+4x+C`
`= (2-3)^3/3-(3(2)^2)/2+4(2)+C = 4`
`= -1/3-6+8+C = 4`
`= 5/3+C = 4`
`C = 7/3`

Answer 2

`f(x)=1/3x^3-9/2x^2+13x-20/3`

Explanation:

`"expand "(x-3)^2" and simplify"`

`f(x)=int(x^2-6x+9-3x+4)dx`

`color(white)(f(x))=int(x^2-9x+13)dx`

`"integrate each term using the "color(blue)"power rule"`

`•color(white)(x)int(ax^n)dx=a/(n+1)x^(n+1);n!=-1`

`rArrf(x)=1/3x^3-9/2x^2+13x+c`

`"where c is the constant of variation"`

`"to find c use the condition "(f(2)=4`

`rArr4=8/3-18+26+crArrc=-20/3`

`rArrf(x)=1/3x^3-9/2x^2+13x-20/3`