What is F(x) = int e^(x-2) - 2x^2 dx if F(0) = 1 ?

1 Answer
Mar 12, 2016

F(x)=e^(x-2)-2/3x^3+1-e^-2

Explanation:

Step 1: Break it Up
Always, always look for ways to simplify a problem before you start solving it. Using the properties of integrals, we can break this big integral up into two smaller ones:
inte^(x-2)-2x^2dx=inte^(x-2)dx-int2x^2dx

Step 2: Solve the Integrals
The first integral is very easy - if you know your exponent rules. e^(x-2) can be rewritten as e^xe^-2, using the sum rule for exponents (e^(a+b)=e^ae^b). That means our new integral is:
inte^xe^-2dx
Because e^-2 is a constant, we can pull it out:
inte^(x-2)=e^-2inte^xdx
=e^-2(e^x+C)=e^(x-2)+C

The second integral is also simple - just some reverse power rule:
int2x^2dx=2/3x^3+C

Step 3: Constant of Integration
Our solution is F(x)=e^(x-2)-2/3x^3+C. We are told F(0)=1; that is to say:
1=e^(0-2)-2/3(0)^3+C
Solving for C gives:
1=e^-2+C
C=1-e^-2

Therefore, F(x)=e^(x-2)-2/3x^3+1-e^-2, or F(x)=e^(x-2)-2/3x^3+0.865.