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Using Cauchy's Integral Formula

Cauchy's integral formula statesand can be used in the following way.

Evaluatewhere is the circle

is entire andis in the interior ofhence

Evaluatewhere is the circle

which in the interior ofis entire hence

Evaluatewhere is the circle

has rootsboth insideso we writeand separate the integrand into partial fractions, obtaining

sin z is entire hence

Evaluatewhere is the circle

lie insidebut -2 is outside. We rewrite the integrand using partial fractions.