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Compute the definite integral: ∫02πecosθcos(sinθ) dθ.\int_{0}^{2\pi} e^{\cos \theta} \cos(\sin\theta) \, d\theta.∫02πecosθcos(sinθ)dθ.
Hint: Consider the function f(t)=∫02πetcosθcos(tsinθ) dθf(t) = \int_{0}^{2\pi} e^{t \cos\theta} \cos(t\sin\theta) \, d\thetaf(t)=∫02πetcosθcos(tsinθ)dθ
and use differentiation under the integral sign.
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