Simple ?

Calculus Level 4

Given that \[ { I }(n)=\int _{ 0 }^{ \frac { \pi }{ 4 } }{ \cos ^{ n }{ x } \, dx },\] if \(I(5) = \frac { a\sqrt { b } }{ c } \), what is \(a + b + c?\)

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Details and Assumptions:

  • You can use \(\int { \cos { ax } } \, dx =\frac { 1 }{ a } \sin { ax } \).
  • \(a, b, c\) are positive integers, \( \gcd(a,c)=1,\) and \(b\) is a square-free integer.
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