IMPROPER INTEGRALS
265
32.42 Investigate
But,
For 0<*<1, l-x
4 = (l-x)(l + x)(l+x
2 )<4(l-x). Hence,
Thus,
32.43 Determine whether
converges.
For
and (Problem 32.24)
converges. Hence
converges.
32.44
Determine whether
cos x dx converges.
Since the latter limit does not exist,
cos x dx is not convergent.
32.45
Evaluate
32.46
Evaluate
Hence, the improper integral has the value 2.
32.47
Show that the region in the first quadrant under the curve y = 1 /(x + I)
2
has a finite area but does not have a
centroid.
However,
Hence, the ^-coordinate of the centroid is infinite.
32.48 For what positive values of p is
convergent?
By Problem 32.26, the latter converges when
Let u = l-x, du=-dx. Then
and only when p < 1.
32.49
Evaluate
Let u = x
2 , du = 2xdx.
Then
| • (ir/2) = ?r/4.