What is f(x) = int e^(x+2)-6x dxf(x)=∫ex+2−6xdx if f(2) = 3 f(2)=3?

1 Answer
Zack M.
Feb 4, 2016

f(x) = e^(x+2) - 3x^2 - 39.6f(x)=ex+2−3x2−39.6

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∫ex+2dx−∫6xdx

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

d/dx e^(x+2) = e^(x+2)d/dx (x+2)ddxex+2=ex+2ddx(x+2)

= e^(x+2)=ex+2

So the integral is;

int e^(x+2) dx = int d/dx e^(x+2) dx ∫ex+2dx=∫ddxex+2dx

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

For the second integral we can use the power rule.

int 6x dx = 3x^2 + C∫6xdx=3x2+C

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

e^(x+2) - 3x^2 + Cex+2−3x2+C

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

e^(2+2) - 3(2)^2 + C = 3e2+2−3(2)2+C=3

e^4-12 + C = 3e4−12+C=3

C = 15-e^4=-39.6C=15−e4=−39.6

So;

f(x) = e^(x+2) - 3x^2 - 39.6f(x)=ex+2−3x2−39.6

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