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Question Number 54573 by Abdo msup. last updated on 07/Feb/19
calculate∫0∞cos(e−x2)1+x2dx
Commented by maxmathsup by imad last updated on 08/Feb/19
letI=∫0∞cos(e−x2)1+x2dx⇒I=12∫−∞+∞cos(e−x2)1+x2dx⇒2I=Re(∫−∞+∞eie−x21+x2dx)letconsiderthecomplexfunctionφ(z)=eie−x21+z2thepolesofφareiand−iresidustheoremgive∫−∞+∞φ(z)dz=2iπRes(φ,i)butφ(z)=eie−z2(z−i)(z+i)⇒Res(φ,i)=eie2i⇒∫−∞+∞φ(z)dz=2iπeie2i=π(cos(e)+isin(e))⇒2I=πcos(e)⇒I=π2cos(e).
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