1can anyone kindly give the solution?
lim 0 to 2Π∫ecosθcos(sinθ)dθ = ?
1) 2Î 2) Î 3) Î 2 4) none
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1708$Here $cos(sin\theta)=Re\left \{ cos(sin\theta)+isin(sin\theta)\right \}=Re(e^{isin(\theta)})$\\\\ So $\int_{0}^{2\pi}e^{cos\theta}\times Re(e^{isin\theta})d\theta=Re\left \{ \int_{0}^{2\pi}e^{cos\theta+isin\theta}d\theta \right\}$\\\\ $Re(\int_{0}^{2\pi}e^{e^{i\theta}}d\theta)=$
1I had already done that but can't move forward...........
1I = ∫ecosθcos(sinθ)dθ = ∫Re(eiθ)dθ
= ∫ {1 + cosθ1 + cos2θ2.1 + cos3θ3.2.1 + ....}dθ
taking limits from 0 to 2Î , only first term exists and hence I = 2Î