What is f(x) = int 3x^2+xe^(x)dxf(x)=∫3x2+xexdx if f(0)=3 f(0)=3?

1 Answer
Gió
Jun 17, 2016

I found: f(x)=x^3+e^x(x-1)+4f(x)=x3+ex(x−1)+4

Explanation:

Let us solve the integral first:
int(3x^2+xe^x)dx=3x^3/3+xe^x-int1e^xdx=∫(3x2+xex)dx=3x33+xex−∫1exdx=
=x^3+xe^x-e^x+c=x3+xex−ex+c

let us use the condition f(0)=3f(0)=3 so that for x=0x=0 we have:

3=0^3+0e^0-e^0+c3=03+0e0−e0+c
3=-1+c3=−1+c

so that:

c=4c=4

finally:
f(x)=x^3+xe^x-e^x+4=x^3+e^x(x-1)+4f(x)=x3+xex−ex+4=x3+ex(x−1)+4

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