Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 32C / c / Solution

For
$$
\operatorname{si}(x)=\int_1^\infty\frac{\sin(xt)}t\,dt,
$$
repeated integration by parts gives
$$
\operatorname{si}(x)\sim
\cos x\sum_{n\geq0}(-1)^n(2n)!x^{-2n-1}
+\sin x\sum_{n\geq0}(-1)^n(2n+1)!x^{-2n-2}.
$$


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 Incoming links: Fixed-lower-limit sine-integral expansion