Question

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

Answer 1

`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`