Example 6.6 Evaluate the following real definite integral:
I ¼
Z 1
À1
dx
1 þ x 3 :
ð6:166Þ
We put
f x
ð Þ
1
1 þ x 3 :
ð6:167Þ
Figure 6.19 depicts a graphical form of f (x). The modulus of f (x) tends to be
infinity at x ¼ À 1. Inflection points are present at x ¼ 0 and
ffiffiffiffiffiffiffi ffi
1=2
3
p
% 0:7937
ð
Þ .
Now, in the former two examples, the isolated singular points (i.e., poles) existed
only in the inside of a closed contour. In the present case, however, a pole is located
on the real axis. Bearing in mind this situation, we wish to estimate the integral
(6.166).
Rewriting (6.166), we have
I ¼
Z 1
À1
dx
1 þ x
ð
Þ x 2 À x þ 1
ð
Þ
or I ¼
Z 1
À1
dz
1 þ z
ð
Þ z 2 À z þ 1
ð
Þ
:
The polynomial of the denominator, i.e., (1 + z)(z
2
À z + 1) has a real root at
z ¼ À 1 and complex roots at z ¼ e
Æiπ=3
¼
1Æ
ffiffi
3
p
i
2
. Therefore,
f z
ð Þ
1
1 þ z
ð
Þ z 2 À z þ 1
ð
Þ
-5
-3
-1
1
3
5
Fig. 6.19 Graphical form
of f x
ð Þ
1
1þx 3 . The modulus
of f(x) tends to infinity at
x ¼ À 1. Inflection points
are present at x ¼ 0 and
ffiffiffiffiffiffiffi ffi
1=2
3
p
% 0:7937
ð
Þ
6.8 Examples of Real Definite Integrals
235