Question

What is `f(x) = int (x+3)^2+3x dx` if `f(5)=2`?

Answer 1

`f(x) = 1/3 x^3 + 9/2 x^2 + 9x - 1183/6`

Explanation:

`f(x) = int ((x+3)^2+3x)dx`

This integral could be solved by doing a `u` substitution where `u=x+3`, OR by expanding out the binomial in the parentheses.

I'll expand out the value in the parentheses for simplicity (and to use less extraneous variables).

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

`f(x) = int(x^2+9x+9)dx`

`f(x) = 1/3 x^3 + 9/2 x^2 + 9x +C`

Plug in the initial condition and solve for `C` (an arbitrary constant):
`f(5) = 2`

`2 = 1/3 (5)^3 + 9/2 (5)^2 + 9(5) +C`

I used a calculator in this step (not necessary)
`C = (-1183)/6 approx -197.1667`

Using this value for `C` in the equation for `f(x)`:

`f(x) = 1/3 x^3 + 9/2 x^2 + 9x - 1183/6`