Question

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

Answer 1

`f(x)=3x^3-23`

Explanation:

The first step is to simplify the expression to be integrated.

`(3x+1)^2-6x-1=9x^2+6x+1-6x-1=9x^2`

`rArrint9x^2dx`

integrate using `color(blue)"power rule for integration"`

`color(orange)"Reminder " color(red)(|bar(ul(color(white)(a/a)color(black)(intax^n dx=a/(n+1)x^(n+1))color(white)(a/a)|)))`

`rArrint9x^2dx=9/3x^3+c=3x^3+c" c is a constant"`

Using f(2) = 1 allows c to be calculated.

`3(2)^3+c=1rArr24+c=1rArrc=-23`

`rArrf(x)=3x^3-23`