In Ramanujan's first letter to G.H. Hardy he defines a function $\phi(n)$ as such
$$ \phi(n) = \int_{0}^{\infty} \frac{\cos n x}{e^{2 \pi \sqrt{x}}-1} d x $$ And then he gives a functional equation
$$ \int_{0}^{\infty} \frac{\sin n x}{e^{2 \pi \sqrt{x}}-1} d x=\phi(n)-\frac{1}{2 n}+\phi\left(\frac{\pi^{2}}{n}\right) \sqrt{\frac{2 \pi^{3}}{n^{3}}}$$ How does one prove this functional equation? I am relatively unfamiliar with the methods one would use to solve this, and therefore do not have a proper attempt to share in this question. Thank you