If I am asked to prove (that is exactly the phrasing, not 'establish') a reduction formula, e.g. for $I_n=\int_0^1 x^n \cosh x~ \mathrm{d} x $ that $I_n=\sinh 1 - n\cosh 1 + n(n-1)I_{n-2}$ where $n\geqslant0$, is it sufficient to just integrate by parts so that \begin{align*} I_n&=x^n \sinh x \bigg|_0^1 - n\int_0^1 x^{n-1}\sinh x~\mathrm{d}x\\&=\sinh1 - n\left[x^{n-1}\cosh x\bigg |_0^1 -(n-1)\int_0^1x^{n-2}\cosh x ~\mathrm{d}x \right]\\&=\sinh 1 -n\cosh1-n(n-1)I_{n-2}\end{align*} or do I have to prove it otherwise by induction?
Thank you.