27. vi
2
vii Differentiate x 3 log x and hence
2x
x 2 1dx
1
logx 11
2
2
log 5 log 2
5
log
2
integrate x 2 log x
28. vi
2
vii Differentiate x 3 log x and hence
2x
x 2 1dx
1
logx 11
2
2
log 5 log 2
5
log
2
integrate x 2 log x
d 3
3 1
x log x x log x 3x 2
dx
x
29. vi
2
vii Differentiate x 3 log x and hence
2x
x 2 1dx
1
logx 11
2
2
log 5 log 2
5
log
2
integrate x 2 log x
d 3
3 1
x log x x log x 3x 2
dx
x
x 2 3 x 2 log x
30. vi
2
vii Differentiate x 3 log x and hence
2x
x 2 1dx
1
logx 11
2
2
log 5 log 2
5
log
2
integrate x 2 log x
d 3
3 1
x log x x log x 3x 2
dx
x
x 2 3 x 2 log x
x
2
3 x 2 log x dx x 3 log x c
31. vi
2
vii Differentiate x 3 log x and hence
2x
x 2 1dx
1
logx 11
2
2
log 5 log 2
5
log
2
integrate x 2 log x
d 3
3 1
x log x x log x 3x 2
dx
x
x 2 3 x 2 log x
x
2
3 x 2 log x dx x 3 log x c
3 x 2 log xdx x 3 log x x 2 dx c
32. vi
2
vii Differentiate x 3 log x and hence
2x
x 2 1dx
1
logx 11
2
2
log 5 log 2
5
log
2
integrate x 2 log x
d 3
3 1
x log x x log x 3x 2
dx
x
x 2 3 x 2 log x
x
2
3 x 2 log x dx x 3 log x c
3 x 2 log xdx x 3 log x x 2 dx c
1 3
1 3
x log xdx 3 x log x 9 x c
2
33. vi
2
vii Differentiate x 3 log x and hence
2x
x 2 1dx
1
logx 11
2
2
log 5 log 2
5
log
2
integrate x 2 log x
d 3
3 1
x log x x log x 3x 2
dx
x
x 2 3 x 2 log x
x
2
3 x 2 log x dx x 3 log x c
3 x 2 log xdx x 3 log x x 2 dx c
1 3
1 3
x log xdx 3 x log x 9 x c
2
Exercise 12D; 1 to 12 ace in all, 14a*
Exercise 12E; 1 to 6 all, 7 to 21 odds, 22abc*, 23*