 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2024

Paper 3

Regularity at zero gives
$$
r^2\phi'(r)=\int_0^r\frac{s^2}{1+s^2},ds
=r-\arctan r.
$$
Integrating once more,
$$
\phi(r)=\frac12\log(1+r^2)+\frac{\arctan r}{r}+C,
$$
where $\arctan r/r$ has removable <limit> one at zero. At $r=1$, the nonconstant terms sum to $\tfrac12\log2+\tfrac\pi4$. Hence
$$
\boxed{
\phi(r)=\frac12\log(1+r^2)+\frac{\arctan r}{r}
-1-\frac12\log2-\frac\pi4}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 3 / 9B / c / iii / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 3 / 9B / c / iii / Solution

 Ancestors (12)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 3 / 9B / c / iii / Solution

 Ancestors (12)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)