Questions in a classroom

Calculate $ \ oint \ limits _ {| z | = 1} \ frac {1} {z + 2} DZ, $ and prove that:

$ \ Int _ {0} ^ {\ PI} \ frac {1 + 2 \ cos \ Theta} {5 + 4 \ cos \ Theta} d \ Theta = 0 $

Evidence: the basic theorem of $ Cauchy-Goursat $: $ \ oint \ limits _ {| z | = 1} \ frac {1} {z + 1} dx = 0. $

Because: $ \ oint \ limits _ {| z | = 1} \ frac1 {z + 1} DZ $

$ = \ Int _ {0} ^ {2 \ PI} \ frac {ie ^ {I \ Theta }}{ e ^ {I \ Theta} + 2} d \ Theta $

$ =-\ Int _ {0} ^ {2 \ PI} \ frac {\ sin \ Theta} {5 + 4 \ cos \ Theta} d \ Theta + I \ int _ {0} ^ {2 \ PI} \ frac {1 + 2 \ cos \ Theta} {5 + 4 \ cos \ Theta} d \ Theta $

Apparently: $ \ int_0 ^ {2 \ PI} \ frac {1 + 2 \ cos \ Theta} {5 + 4 \ cos \ Theta} d \ Theta = 0 $

Order $ \ theta-\ Pi = T $

Available: $ \ int_0 ^ {2 \ PI} \ frac {1 + 2 \ cos \ Theta} {5 + 4 \ cos \ Theta} d \ Theta = 2 \ int_0 ^ {\ pi} \ frac {1 + 2 \ cos \ Theta} {5 + 4 \ cos \ Theta} d \ Theta = 0 $

So: $ \ int_0 ^ {\ PI} \ frac {1 + 2 \ cos \ Theta} {5 + 4 \ cos \ Theta} d \ Theta = 0 $

Questions in a classroom