Integral of $xe^y \,dy$

Question

I tried integration by parts but that leaves me with $xe^y$ - integral of $e^y \,dx$.

As far as I know that should leave me with $xe^y - xe^y$ which is $0$. My book says its $xe^y$.

It doesn't seem like substitution is going to work because I have two variables.

If you can explain this to me I'd appreciate it.

Answer 1

Of course it is $xe^y$:$$\int xe^y\,\mathrm dy=x\int e^y\,\mathrm dy=xe^y.$$In this problem, $x$ is a constant.

Answer 2

If $x$ does'nt depend on $y$,

$$\int xe^ydy=xe^y+f(x)$$ $f(x)$ is an arbitrary function.

A particular case is : $f(x)=C$ , then $\int xe^ydy=xe^y+C$

Answer 3

If $x,y$ are dependent on each other, then integration by parts tells you $$ \int xe^y\;dy = x e^y - \int e^y\;dx $$ and the new integral is not zero.