Page 58 - Calculus with Complex Numbers

E v a I u a t i o n o f f i n i t e

r e a I i n t e g r a I s

As a lirst application of the residue theorem (see Section 4.7) we describe a method
for evaluating a certain class of real integrals over a linite interval.

Example 1 Consider the integral

1:: dt
0 5 + 4 cos t '
W e can transform this integral into a contour integral round the unit circle by
making the substitution z = eit W e have dz = i eitdt = izdt which gives
dt = dz/iz. We also have

Therefore we get

4
j 5 dz 1 1 dz
z:c o!r os t - j iz 5 + 2tc + 1/c) - 'j- Jy 2c2 + 5c + 2 '

+
c

where y is the unit circle.
W e now evaluate this contour integral using the residue theorem. Observe that
2z2 + 5 . z + 2 = (2c + 1) (c + 2), therefore the singularities of the integrand occur
at z = -2, - 1/2 (Figure 5. 1). Of these only z = - 1/2 is inside y, where the
residue is
1 1 1
= ,
-/ 4 .: + 5 =-1yz 3i
using the method of differentiating the denominator (see Section 4.8).
Hence we have

y
U
U
+

:

2
.
'
+
4

1 5 o J 2 5 1 l:,r
1:: dt 1 dz .: + 2 '''U CK i X j/ = 3 '

t
'
'
c
s