A calculus problem by Priyanshu Mishra

Calculus Level 4

If

\[\large\ \int _{ 0 }^{ \pi }{ { e }^{ \left| \cos { x } \right| }\left\{ 2\sin { \left( \frac { 1 }{ 2 } \cos { x } \right) } + 3\cos { \left( \frac { 1 }{ 2 } \cos { x } \right) } \right\} \sin { x } dx } = \frac { m }{ n } \left[ e\left( \cos { \frac { 1 }{ a } + \frac { 1 }{ b } \sin { \frac { 1 }{ c } } } \right) - d \right] \]

where \(m, n, a, b, c, d\) are all natural numbers

Find the value of \[\large\ m + n + a + b + c + d\].

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