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

1 Answer
Jun 24, 2016

The answer consists in calculating the primitive f (antiderivative) of the given function, and then evaluating the constant for which the primitive f(2) = 3

Explanation:

The integral distributes with respect to the sum, so:
int (e^(x+2) + x) dx = int e^(x+2) dx + int x dx

So, solving the first one:
int e^(x+2) dx = int e^x e^2 dx = e^2 int e^x dx= e^2 e^x + C_1 where C_1 is an unknown constant.

Similarly, the second one is:
int x dx = x^2/2 + C_2

So the antiderivative of e^(x+2) + x is:
f(x) = e^2 e^x + x^2/2 + C, where C is a constant.

Now we have to solve for f(2) = 3:
e^2 e^2 + 2^2/2 + C = 3, that is:
e^4 + 2 + C = 3, thus:
e^4 + C = 1, and then
C = 1-e^4.

The final answer is then:
f(x) = e^2 e^x + x^2/2 + (1-e^4)