Page 147 - Calculus Demystified

CHAPTER 5
134

Fig. 5.3 Indeterminate Forms

EXAMPLE 5.14
Evaluate the integral
8

4(8 − x) −1/3 dx.
2
SOLUTION
The integral

8
4(8 − x) −1/3 dx
2
is an improper integral with infinite integrand at 8. According to the definition,
the value of this integral is

8−(
lim 4(8 − x) −1/3 dx,
(→0 + 2
provided the limit exists. Since the integrand is continuous on the interval
[2, 8 − (], we may calculate this last integral directly. We have

8−(
2/3 2/3
lim − 6(8 − x) 2/3 = lim −6 ( − 6 .
(→0 + 2 (→0 +
This limit is easy to evaluate: it equals 6 5/3 . We conclude that the integral is
convergent and

8
4(8 − x) −1/3 dx = 6 5/3 .
2