The integration ratio

Calculus Level 5

In=0πex(sinx)ndx\large{I_{n}=\displaystyle \int^{\pi}_{0} e^{x}(\sin x)^{n} \, dx}

Define InI_n as the integral above. If I5I1=ab\large{\dfrac{I_{5}}{I_{1}}=\dfrac{a}{b}} where a,ba,b are coprime integers, find a+ba+b.

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