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

1 Answer
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