0∫2π2+cosθdθ=0∫2π4+eiθ+e−iθ2dθ=0∫2π4eiθ+e2iθ+12eiθdθ=
=i20∫2πe2iθ+4eiθ+1deiθ=i2∣z∣=1∮z2+4z+1dz=i2∣z∣=1∮(z+2+3)(z+2−3)dz
Only one pole, z=−2+3, is inside the contour {∣z∣=1}, therefore, by the residue theorem we have
0∫2π2+cosθdθ=i22πiresz=−2+3(z+2+3)(z+2−3)1=
=4πz→−2+3limz+2+31=32π
Answer. 32π
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