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

1 Answer
May 20, 2017

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