What is f(x) = int 3x^3-2x+xe^x dx if f(1) = 3 ?

1 Answer
Mar 5, 2017

The answer is =3/4x^4-2/2x^2+e^x(x-1)+13/4

Explanation:

We need

intx^ndx=x^(n+1)/(n+1)+C(x!=-1)

We start, by calculating

intxe^xdx

by integration by parts

intuv'dx=uv-intu'vdx

u=x, =>, u'=1

v'=e^x, =>, v=e^x

Therefore,

intxe^xdx=xe^x-inte^xdx=xe^x-e^x

So,

f(x)=int(3x^3-2x+xe^x)dx

=3intx^3dx-2intxdx+intxe^xdx

=3/4x^4-2/2x^2+e^x(x-1)+C

f(1)=3/4-1+0+C=3

C=3+1/4=13/4

Therefore,

f(x)=3/4x^4-2/2x^2+e^x(x-1)+13/4