What is f(x) = int x^2-2xe^(x)dxf(x)=∫x2−2xexdx if f(0)=-2 f(0)=−2?
1 Answer
Explanation:
Separate the integral.
f(x) = int x^2dx - int 2xe^xdxf(x)=∫x2dx−∫2xexdx
The first integral is
f(x) = 1/3x^3 - int 2xe^xdxf(x)=13x3−∫2xexdx
We will use integration by parts for the last integral. We let
int 2xe^xdx = 2x(e^x) - int 2e^x∫2xexdx=2x(ex)−∫2ex
int2xe^xdx = 2xe^x - 2inte^x∫2xexdx=2xex−2∫ex
int2xe^x = 2xe^x - 2e^x∫2xex=2xex−2ex
int2xe^x = 2e^x(x - 1)∫2xex=2ex(x−1)
We put the integral back together to find
f(x) = 1/3x^3 - 2e^x(x - 1) + Cf(x)=13x3−2ex(x−1)+C
We must now find the value of
-2 = 1/3(0)^3 - 2e^0(0 - 1) + C−2=13(0)3−2e0(0−1)+C
-2 = 0 + 2 + C−2=0+2+C
C = -4C=−4
This means that
Hopefully this helps!
