The integration ratio

Calculus Level 5

\[\large{I_{n}=\displaystyle \int^{\pi}_{0} e^{x}(\sin x)^{n} \, dx}\]

Define \(I_n\) as the integral above. If \(\large{\dfrac{I_{5}}{I_{1}}=\dfrac{a}{b}}\) where \(a,b\) are coprime integers, find \(a+b\).

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