Cauchy's integral formula states
and can be used in the following way.
Evaluate
where 
is the circle
is entire and
is in the interior ofhence 
Evaluate
where is the circle
which in the interior of
is entire hence
Evaluate
where is the circle
has roots
both insideso we write
and separate the integrand into partial fractions, obtaining
sin z is entire hence
Evaluate
where is the circle
lie insidebut -2 is outside. We rewrite the integrand using partial fractions.
