Which of the following discoveries resulted in a Nobel Prize?

If $0 < y < 2^{1/3}$ and $x(y^3 - 1) = 1$,then $\frac{2}{x} + \frac{2}{3x^3} + \frac{2}{5x^5} + \dots$ is equal to:

For a crystal,the angle of diffraction $(2 \theta)$ is $90^{\circ}$ and the second order line has a $d$ value of $2.28 \ \mathring{A}$. The wavelength (in $\mathring{A}$) of $X$-rays used for Bragg's diffraction is

The height of water in a tank is $H$. The range of the liquid emerging out from a hole in the wall of the tank at a depth $\frac{3H}{4}$ from the upper surface of water will be

$\int \frac{dx}{\sqrt{x}(x+9)}$ is equal to

If $y$ is a function of $x$ and $\log(x+y) = 2xy$,then the value of $y'(0)$ is