Question

What is `f(x) = int xe^x-xsqrt(x^2+2)dx` if `f(0)=-2`?

Answer 1

`f(x)`=`int xe^x*dx`-`int x*sqrt(x^2+2)*dx`

=`x*e^x`-`int e^x*dx`-`1/3*(x^2+2)^(3/2)+C`

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

After imposing `f(0)=-2` condition,

`(0-1)*e^0-1/3*2^(3/2)+C=-2`

`C=(2sqrt(2))/3-1`

Hence `f(x)`=`(x-1)e^x`-`1/3*(x^2+2)^(3/2)+(2sqrt(2))/3-1`

Explanation:

1) I took integral right side.

2) I imposed `f(0)=-2` condition for finding `C`.