Wanna integrate by parts 10 times? (My ninth integral problem)

Calculus Level 5

\[ \int_0^{\pi/2} x^{10} \cos x \, dx \]

If the value of the integral above can be represented in the incredibly long form of

\[ -a + b \pi^{2} - c \pi^{4} + \dfrac{d}{e} \pi^{6} - \dfrac{f}{g} \pi^{8} + \dfrac{\pi^{10}}{h} ,\]

where \(a,b,c,d,e,f,g,h\) are positive integers with \(\gcd(d,e) = \gcd(f,g) = 1\), find \(a+b+c+d+e+f+g+h\).

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