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I-0-pi-e-acos-t-cos-asin-t-dt-




Question Number 190754 by universe last updated on 10/Apr/23
      I   =    ∫_0 ^( π) e^(acos t)  cos (asin  t)dt
$$\:\:\:\:\:\:{I}\:\:\:=\:\:\:\:\int_{\mathrm{0}} ^{\:\pi} {e}^{{a}\mathrm{cos}\:{t}} \:\mathrm{cos}\:\left({a}\mathrm{sin}\:\:{t}\right){dt} \\ $$
Answered by namphamduc last updated on 11/Apr/23
I=∫_0 ^π e^(acos(t)) cos(asin(t))dt  =(1/2)ℜ∫_(−π) ^π e^(acos(t)) e^(iasin(t)) dt=(1/2)ℜ∫_(−π) ^π e^(acos(t)+iasin(t)) dt  =(1/2)ℜ∫_(−π) ^π e^(ae^(it) ) dt,z=e^(it) ⇒dz=ie^(it) dt⇒dt=−i(dz/z)  ⇒I=(1/2)ℜ(−i∫_(∣z∣=1) (e^(az) /z)dz)=(1/2)ℜ(−i.2πi.Res((e^(az) /z),z=0))=π
$${I}=\int_{\mathrm{0}} ^{\pi} {e}^{{a}\mathrm{cos}\left({t}\right)} \mathrm{cos}\left({a}\mathrm{sin}\left({t}\right)\right){dt} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\Re\int_{−\pi} ^{\pi} {e}^{{a}\mathrm{cos}\left({t}\right)} {e}^{{ia}\mathrm{sin}\left({t}\right)} {dt}=\frac{\mathrm{1}}{\mathrm{2}}\Re\int_{−\pi} ^{\pi} {e}^{{a}\mathrm{cos}\left({t}\right)+{ia}\mathrm{sin}\left({t}\right)} {dt} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\Re\int_{−\pi} ^{\pi} {e}^{{ae}^{{it}} } {dt},{z}={e}^{{it}} \Rightarrow{dz}={ie}^{{it}} {dt}\Rightarrow{dt}=−{i}\frac{{dz}}{{z}} \\ $$$$\Rightarrow{I}=\frac{\mathrm{1}}{\mathrm{2}}\Re\left(−{i}\int_{\mid{z}\mid=\mathrm{1}} \frac{{e}^{{az}} }{{z}}{dz}\right)=\frac{\mathrm{1}}{\mathrm{2}}\Re\left(−{i}.\mathrm{2}\pi{i}.\mathrm{Res}\left(\frac{{e}^{{az}} }{{z}},{z}=\mathrm{0}\right)\right)=\pi \\ $$

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