Integrating exponentials and complex numbers: Why does this equality hold?

Question

I'm looking at a physics problem and I'm having trouble to understand why the following equality holds:

$\int{p e^{\frac{i}{h}xp}dp}=\int{\frac{h}{i}\frac{d}{dx}[e^{\frac{i}{h}xp}]dp}$

I would be grateful if someone could clarify the steps in between. Also, note that i stands for the imaginary unit and h is just a constant. Thanks.

Answer 1

In general, for any constant $a$, $$\frac{d}{dx}e^{ax}=ae^{ax}.$$ Now, in $\frac{d}{dx}e^{\frac ihxp}$, $p$ is merely a constant (since we're differentiating with respect to $x$). Can you proceed from here?

Answer 2

By Chain Rule $\frac {d} {dx} e^{\frac i h xp}=e^{\frac i h xp} \frac d {dx} \frac i h xp=\frac {ip} h e^{\frac i h xp}$. Now just multiply by $\frac h i$