Q:

Why does this integral vanish while doing integration by parts?

Consider $$\int_{-\infty}^{+\infty} x \dfrac{\partial}{\partial x}\left(\Psi^*\dfrac{\partial \Psi}{\partial x} - \Psi\dfrac{\partial \Psi^*}{\partial x}\right) dx $$
If I apply integraton by parts here by bringing in the x inside the derivative then its $$\int_{-\infty}^{+\infty} \dfrac{\partial}{\partial x}\left(x\cdot\left(\Psi^*\dfrac{\partial \Psi}{\partial x} - \Psi\dfrac{\partial \Psi^*}{\partial x}\right)\right) dx $$ and taking $f = x$ and $g' = (\Psi^*\dfrac{\partial \Psi}{\partial x} - \Psi\dfrac{\partial \Psi^*}{\partial x})$ the integral should be:
$$-\int_{-\infty}^{+\infty} \dfrac{\partial x}{\partial x}\left(\int \left(\Psi^*\dfrac{\partial \Psi}{\partial x} - \Psi\dfrac{\partial \Psi^*}{\partial x}\right) dx\right)dx$$. We know that $\dfrac{\partial x}{\partial x} = 1$ so integral becomes 
$$-\iint_{-\infty}^{+\infty}  \left(\Psi^*\dfrac{\partial \Psi}{\partial x} - \Psi\dfrac{\partial \Psi^*}{\partial x}\right) dxdx$$
Why isn't this right? The answer is just:$$-\int_{-\infty}^{+\infty}  \left(\Psi^*\dfrac{\partial \Psi}{\partial x} - \Psi\dfrac{\partial \Psi^*}{\partial x}\right) dx$$
where did the other integral vanish?
Also, I ignored the boundary term because $\Psi \to 0$ at $\pm\infty$.

A:

You are not applying integration by parts correctly. You take 
$$
g' = \Bigl(\Psi^*\dfrac{\partial \Psi}{\partial x} - \Psi\,\dfrac{\partial \Psi^*}{\partial x}\Bigr),
$$
and it should be
$$
g= \Bigl(\Psi^*\dfrac{\partial \Psi}{\partial x} - \Psi\,\dfrac{\partial \Psi^*}{\partial x}\Bigr).
$$
Then
$$
\int_{-\infty}^\infty x\,\frac{\partial}{\partial x}g(x)\,dx=\text{boundary term}-\int_{-\infty}^\infty g(x)\,dx.
$$

