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

1 Answer
Zack M.
Feb 4, 2016

#f(x) = e^(x+2) - 3x^2 - 39.6#

Explanation:

We need to start by solving the integral, then plugging in the known values to finish. Notice that inside the integral is a subtraction symbol, which tells us that we can split this integral into two parts.

#int e^(x+2) dx - int 6x dx#

For the first one, start by using the chain rule to find the derivative of #e^(x+2)#.

#d/dx e^(x+2) = e^(x+2)d/dx (x+2)#

#= e^(x+2)#

So the integral is;

#int e^(x+2) dx = int d/dx e^(x+2) dx #

#= e^(x+2) + C#

For the second integral we can use the power rule.

#int 6x dx = 3x^2 + C#

Putting everything together and combining the constants into one term, the general solution for our integral is;

#e^(x+2) - 3x^2 + C#

We are given the point #(2,3)#, so we can plug this in to solve for #C#.

#e^(2+2) - 3(2)^2 + C = 3#

#e^4-12 + C = 3#

#C = 15-e^4=-39.6#

So;

#f(x) = e^(x+2) - 3x^2 - 39.6#

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