How to solve combinations of differentials and integrals? In the first chapter of Nearing's book "Mathematical tools for physics" (available online) I encountered an interesting combination of differentials and integrals - which I don't fully understand:
(a) $$\frac{\mathrm{d} }{\mathrm{d}\alpha}\int_{-\infty}^{\infty}e^{-\alpha x^2}dx=-\int_{-\infty}^{\infty}x^2e^{-\alpha x^2}dx$$
(b) $$\frac{\mathrm{d} }{\mathrm{d} x}\int_{0}^{x}e^{-x t^2}dt=e^{-x^{3}}-\int_{0}^{x}t^2e^{-x t^2}dt$$
(c) $$\frac{\mathrm{d} }{\mathrm{d} x}\int_{x^2}^{\,\sin x}e^{x t^2}dt=e^{x\, \sin^2 x} \,\cos x-e^{x^{5}}2x+\int_{x^2}^{\,\sin x}t^2e^{x t^2}dt$$
I can't see in which order you have to do which rules to arrive at the solutions. Could anyone please give me the steps in between? Thank you!
 A: It's a Leibniz integral rule
 -see e.g. in wikipedia by the link.
A: Basically all those identities are just using differentiation under the integral sign. For example for the first one, you have a function of two variables $f(x, \alpha) : = e^{-\alpha x^2}$ and then the identity is obtained by interchanging the order of the derivative with the integral
$$\frac{\mathrm d}{ \mathrm d \alpha} \int_{-\infty}^{\infty} e^{-\alpha x^2} \, \mathrm{dx} =\frac{\mathrm d}{ \mathrm d \alpha} \int_{-\infty}^{\infty} f(x, \alpha) \, \mathrm{dx} = \int_{-\infty}^{\infty} \frac{\partial f(x, \alpha)}{\partial \alpha} \, \mathrm dx $$
$$ = \int_{-\infty}^{\infty} \frac{\partial}{\partial \alpha} (e^{-\alpha x^2}) \, \mathrm dx = \int_{-\infty}^{\infty} -x^2 e^{-\alpha x^2} \, \mathrm{dx}$$
Of course some conditions must be satisfied for this in order to work, but you can check what those conditions are in the previous link. You can find this treated in most analysis books. For example it is treated in chapter 9 of Rudin's Principles of Mathematical Analysis under the section "Differentiation of Integrals".
