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Calculus Level 4

0πecosθcos(sinθ)dθ \large \int_0^\pi e^{\cos \theta} \cos (\sin \theta) d \theta

The above definite integral evaluates to a number of the form aπb+k a \pi^b + k where a a , b b and k k are integers. Find the sum of the digits of the number (a+b+k)12 (a + b + k)^{12} .

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