calculate-cos-2x-dx-x-4-x-2-1- Tinku Tara June 3, 2023 Integration 0 Comments FacebookTweetPin Question Number 136034 by mathmax by abdo last updated on 18/Mar/21 calculate∫−∞+∞cos(2x)dxx4+x2+1 Answered by mathmax by abdo last updated on 19/Mar/21 Φ=∫−∞+∞cos(2x)x4+x2+1dx⇒Φ=Re(∫−∞+∞e2ixx4+x2+1dx)letφ(z)=e2izz4+z2+1polesofφ?z4+z2+1=0⇒u2+u+1=0(z2=u)Δ=−3⇒u1=−1+i32=e2iπ3andu2=e−2iπ3⇒φ(z)=e2iz(z2−e2iπ3)(z2−e−2iπ3)=e2iz(z−eiπ3)(z+eiπ3)(z−e−iπ3)(z+eiπ3)∫−∞+∞φ(z)dz=2iπ(Res(φ,eiπ3)+Res(φ,−e−iπ3))Res(φ,eiπ3)=e2ieiπ32eiπ3(2isin(2π3))=e2i(12+i32)4i.32e−iπ3=12i3e−3.ei−iπ3=12i3e−3.e2iπ3Res(φ,−e−iπ3)=e−2ie−iπ3−2e−iπ3(−2isin(2π3))=e−2i(12−i32)4i.32.eiπ3=12i3.e−3.e−i+iπ3=12i3e−3.e−2iπ3⇒∫Rφ(z)dz=2iπ.12i3e−3{e2iπ3+e−2iπ3}=π3e−3.(2cos(2π3))=−π3e−3⇒∫−∞+∞cos(2x)x4+x2+1dx=−π3e−3 Terms of Service Privacy Policy Contact: info@tinkutara.com FacebookTweetPin Post navigation Previous Previous post: Question-70497Next Next post: Question-70499 Leave a Reply Cancel replyYour email address will not be published. Required fields are marked *Comment * Name * Save my name, email, and website in this browser for the next time I comment.