234
CHAPTER 28
28.19
28.20
28.21
28.22
28.23
28.24
28.25
28.26
28.27
28.28
28.29
Let 3x = -y and use Problem 28.1:
(9x
2 -6x + 2)+C.
Then J x
2 tan
t xdx=^x
3 tan
l xA simple substitution works: let u = 1 + x , du = 2x dx.
Let 9? be the region bounded by the curve y = In x, the *-axis, and the line x = e. Find the area of 91.
Then
Let
M = In x.
dv = x
2 dx,
du = (1 Ix) dx.
Find the volume of the solid obtained by revolving the region $1 of Problem 28.25 about the x-axis.
By the disk formula,
2+ 2)-2] =w(e-2).
Find the volume of the solid obtained by revolving the region 9/1 of Problem 28.25 about the y-axis.
We use the cylindrical shell formula:
Let Sfc be the region bounded by the curve y = In x/x, the *-axis, and the line x = e. Find the area of £%.
Find the volume of the solid obtained by revolving the region 3? of Problem 28.28 about the y-axis.
By the cylindrical shell formula,
J x
1 In x dx.
Let M = ln*, dv = (l/x
2 ) dx, du = (l/x)dx, v = -l/x. Then
JjrVdr.
(y
2 + 2>> + 2)+C =
J je
2 tan ' x dc.
Let M = tan~'j«:, dv = x
2 dx, du = [1/(1 + x
2 )] dx,
ln(l + jO
ln(l + jc
2 )+C.
fln(x
2 + l)dx.
Let u = ln(*
2 + l),
dv = dx, du = [2x/(x
2 + 1)] dx, v = x. So J ln(x
2 + 1) dx = xln (x
2 + 1) -
dx = x In (x
2 + 1) - 2(x - tan"
1 x) + C = x In (x
2 + 1) - 2x +
2tan~
1 ;e+ C.
By Problem 28.16, v = 7rx[(ln x)
2 - 2 In x + 2}