Solving Complex Integrals Using the Residue Theorem on Circular Contours

[nickolas2730]

1. Evaluate ∫_C (z)/z²+9 dz , where C is the circle │z-2i│=4.

what i have done so far is :

z(t) = 2i + 4e^it
z'(t) = 4ie^it
f(z(t)) = 4ie^it/(4ie^it)²+9

∫ (4ie^it/(4ie^it)²+9) (4ie^it) dt

intergrate from 0->2pi

but i don't know how to solve this intergral, can anyone help?

2. ∫_c cos(z)/(z-1)^3(z-5)^2 dz , where C is the circle │z-4│=2.

this z'(t) = 0
so , is this intergral equal 0?
since f(z(t))(z'(t)) = 0


Thanks

 

[obafgkmrns]

Shouldn't you be using the residue theorem to evaluate the integrals?

(Note that (1) has poles at ±i3, and (2) has poles at 1 & 5. Which of those lie within the given contours?)