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

1 Answer
Oct 3, 2017

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.