(i) Let $\phi$ satisfy $\phi=f+\lambda K \phi$ in $[0,1]$ where $f(x)=1(0 \leqslant x \leqslant 1)$ and $(K \phi)(x)=\int_0^x t \log (t / x) \phi(t) \mathrm{d} t \quad(0 \leqslant x \leqslant 1)$. Show by induction that $\left(K^n f\right)(x)=\left(-\frac{1}{4} x^2\right)^n /(n !)^2(n \in \mathbb{N})$ and hence prove that
$$
\phi(x)=1+\sum_{n=1}^{\infty}\left(-\frac{1}{4} i x^2\right)^n /(n !)^2 \quad(0 \leqslant x \leqslant 1)
$$
satisfies the given integral equation. Show also that, with $\lambda=1$, $\phi(x)=1-\frac{1}{4} x^2+\frac{1}{64} x^4+\epsilon(x)(0 \leqslant x \leqslant 1)$ where $\int_0^1|\epsilon(x)|^2 \mathrm{~d} x<1.4 \times 10^{-4}$.
(ii) Let $\phi=f+\lambda K \phi$ in $(0,1]$ where $f(x)=\log x \quad(0<x \leqslant 1)$ and $K$ is as defined in (i). Show that
$$
\left(K^n f\right)(x)=\left(-\frac{1}{4} x^2\right)^n\left\{\log x-1-\frac{1}{2}-\cdots-\frac{1}{n}\right\} /(n !)^2 \quad(n \in \mathbb{N})
$$
and deduce the solution of the integral equation. Show that, with $\lambda=1$,
$$
\phi(x)=\left(1-\frac{x^2}{4}+\frac{x^4}{64}\right) \log x+\frac{x^2}{4}-\frac{3 x^4}{128}+\epsilon(x) \quad(0<x \leqslant 1),
$$
where $\int_0^1|\epsilon(x)|^2 \mathrm{~d} x<2.7 \times 10^{-4}$.