Question

What is `f(x) = int xe^x` if `f(2)=3`?

Answer 1

`f(x)=xe^x-e^x+3-e^2`

Explanation:

`f(x)=intxe^xdx, f(2)=3`

we use integration by parts

`f(x)=intu(dv)/(dx)dx=uv-intv(du)/(dx)dx`

in this case

`u=x=>(du)/(dx)=1`

`(dv)/(dx)=e^x=>v=e^x`

`:.f(x)=xe^x-inte^xdx`

`f(x)=xe^x-e^x+c`

`f(2)=3`

`:. f(2)=3=2e^2-e^2+c`

`c=3-e^2`

`f(x)=xe^x-e^x+3-e^2`