Page 60 - Calculus with Complex Numbers

Ffgure 5.2

Example 3 Consider the integral

2x
cos
4 t dt
.
0
The substitution z = eit gives

2x
/-

where y is the unit circle.
Observe that the integrand is already a Laurent expansioa indicating that there
is a pole of order 5 at z = 0, and that the residue there is 6/ 16ï = 3/8/ .
Hence we have

z,r (3 3a.
cos4 t dt = ln'i x = .
0 W 4

Example 4 Consider the integral

2x
sin lt cos 3/ dt.
0