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

Question

Charles Arocha · Accepted Answer

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

#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#

Charles Anctil · Answer

undefined To find $ f(x) = \int (x+3)^2+3x \, dx $ if $ f(5) = 2 $, we need to evaluate the definite integral of the given function and then use the given condition to solve for an unknown constant.

Given:
$$ f(x) = \int (x+3)^2+3x \, dx $$

First, integrate the function:
$$ f(x) = \int (x^2 + 6x + 9 + 3x) \, dx $$
$$ = \int (x^2 + 9x + 9) \, dx $$
$$ = \frac{1}{3}x^3 + \frac{9}{2}x^2 + 9x + C $$

Given that $ f(5) = 2 $, we can use this information to find the constant $ C $:
$$ f(5) = \frac{1}{3}(5)^3 + \frac{9}{2}(5)^2 + 9(5) + C = 2 $$

$$ \frac{125}{3} + \frac{225}{2} + 45 + C = 2 $$

$$ C = 2 - \left(\frac{125}{3} + \frac{225}{2} + 45\right) $$
$$ C = 2 - \left(\frac{125}{3} + \frac{225}{2} + \frac{135}{3}\right) $$
$$ C = 2 - \left(\frac{125 + 675 + 270}{6}\right) $$
$$ C = 2 - \left(\frac{1070}{6}\right) $$
$$ C = 2 - \frac{535}{3} $$
$$ C = \frac{6}{3} - \frac{535}{3} $$
$$ C = \frac{-529}{3} $$

Thus, the function $ f(x) = \int (x+3)^2+3x \, dx $ with $ f(5) = 2 $ is:
$$ f(x) = \frac{1}{3}x^3 + \frac{9}{2}x^2 + 9x - \frac{529}{3} $$

Cameron Alexander · Answer

undefined The integral $ \int (x+3)^2+3x \, dx $ is equal to $ \frac{(x+3)^3}{3} + \frac{3x^2}{2} + C $, where $ C $ is the constant of integration. Given that $ f(5) = 2 $, we can solve for $ C $ and then evaluate $ f(x) $ at $ x = 5 $. Solving for $ C $, we have $ \frac{(5+3)^3}{3} + \frac{3(5)^2}{2} + C = 2 $, which simplifies to $ \frac{512}{3} + \frac{75}{2} + C = 2 $. Solving for $ C $, we get $ C = 2 - \frac{512}{3} - \frac{75}{2} $. Substituting this value of $ C $ back into the expression for $ f(x) $, we can evaluate $ f(5) $ to find the value of the integral.