Simple ?

Calculus Level 4

Given that

I(n)=0π4cosnxdx, { I }(n)=\int _{ 0 }^{ \frac { \pi }{ 4 } }{ \cos ^{ n }{ x } \, dx },

if I(5)=abcI(5) = \frac { a\sqrt { b } }{ c } , what is a+b+c?a + b + c?

Details and Assumptions:

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