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Calculus Level 3

\[\large \int_{0}^{\pi /2} e^x\sin x\cos x \, dx\]

If the above integral can be expressed as\[ \large \dfrac{a+{e^{{b\pi} / {c}}}}{d} , \] where \(a,b,c\) and \(d\) are positive integers with \(b\) and \(c\) coprime, find the value of \(a+b+c+d\).

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