There Is No Exact Form, Right?

Calculus Level 5

Given

\[ 34 \times \frac {\displaystyle{ \int_0^{\pi} (\sin x)^{3\sqrt{2} + 3} dx} }{ \displaystyle{\int_0^{\pi /2} (\cos x)^{3\sqrt{2} - 3} dx }} = a \sqrt{2} + b \]

for integers \(a\) and \(b,\) find the value of \(a+b.\)

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