Problem with $\int (1+ye^{xy}) \text{dx}$

Question

When I compute this integral on my TI-89, I get an answer of $x(e^{xy}\cdot y+1 + C)$.

When I use the Internet, I get an answer of $e^{xy} + x+C$.

Are these integrals identical? I can follow the procedure done on the website, but I'm not sure why my calculator isn't giving the same result.

Thanks.

Answer 1

The internet integral is correct.

At first, I thought that the calculator was thinking $xy$ was a separate variable, not the product of $x$ and $y$, but then the $Cx$ should be $C$. I am not quite sure why the calculator is giving the answer it does.

Answer 2

When you take derivative of $$e^{xy} + x+C$$ you get $$ ye^{xy }+1$$

Thus this answer is definitely correct.

The other one does not result in the correct answer.