4 Let \(I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { n } } \mathrm {~d} x\). Prove that, for every positive integer \(n\),
$$2 n I _ { n + 1 } = 2 ^ { - n } + ( 2 n - 1 ) I _ { n }$$
Given that \(I _ { 1 } = \frac { 1 } { 4 } \pi\), find the exact value of \(I _ { 3 }\).
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4 Let $I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { n } } \mathrm {~d} x$. Prove that, for every positive integer $n$,
$$2 n I _ { n + 1 } = 2 ^ { - n } + ( 2 n - 1 ) I _ { n }$$
Given that $I _ { 1 } = \frac { 1 } { 4 } \pi$, find the exact value of $I _ { 3 }$.
\hfill \mbox{\textit{CAIE FP1 2013 Q4}}