# Definite integral with summation

Calculus Level 4

If

$\ { I }_{ n }=\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { x }^{ n }\cos { x } dx }$

Then

$\sum _{ n=2 }^{ \infty }{ \left( \frac { { I }_{ n } }{ n! } +\frac { { I }_{ n-2 } }{ \left( n-2 \right) ! } \right) }$

is of the form $${ e }^{ a }-b-1$$ , find the value of $$a-b$$.

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