438 Chapter 7 Iii The Riemann Integral(g)^11 -d^1 x^15 1 12 1
x3 (h)^2 (x - 3)2 dx (i) -2 --dx x - l
-1(j)^12 dx
-1 x2 +ex(k) 11 exdx
0 Vx(1) 1
1
x lnxdx1 "1
2
(m) sinx -YX dx
0 x(n)^11 -^1
1
-dx
0 x nx- Without using the arcsine function, use the comparison test to prove that
1
1
h converges.- Determine the convergence or divergence of 1
1
x(l~xx) 2.- Prove that lim ;·c ( 2 x ) 2 dx = 0. Does this imply that
c-->1- -c X -l
1: (x 2 ~ l) 2 dx converges to 0? Graph the function f(x) = (x 2 ~ l) 2
over the interval (-1, 1) and explain what is going on.IMPROPER INTEGRALS OF TYPE II
Definition 7.8.9 Suppose a E IR and Vb > a, f is integrable on [a, b]. Then
we call J,+
00
a f an improper integral of type II. If b-->+oo lim t a f exists, we say
that fa+oo f converges, and write
f,+
00
f = lim t f.
a b-->+oo aOtherwise, we say that fa+oo f diverges.
Definition 7 .8. 10 Suppose b E IR and Va < b, f is integrable on [a, b]. Then
we call J~ oo f an improper integral of type II. If a-t-oo lim t a f exists, we say
that J~ 00 f converges, and write
J~ oo f = a-t-oo lim t a f.
Otherwise, we say that J~ 00 f diverges.