27. 2
vi 2x
x 2 1dx vii Differentiate x 3 log x and hence
1
integrate x 2 log x
logx 11
2 2
log 5 log 2
log
5
2
28. 2
vi 2x
x 2 1dx vii Differentiate x 3 log x and hence
1
integrate x 2 log x
logx 11
2
x log x x log x 3x 2
3 1
2
d 3
dx x
log 5 log 2
log
5
2
29. 2
vi 2x
x 2 1dx vii Differentiate x 3 log x and hence
1
integrate x 2 log x
logx 11
2
x log x x log x 3x 2
3 1
2
d 3
dx x
log 5 log 2
x 2 3 x 2 log x
log
5
2
30. 2
vi 2x
x 2 1dx vii Differentiate x 3 log x and hence
1
integrate x 2 log x
logx 11
2
x log x x log x 3x 2
3 1
2
d 3
dx x
log 5 log 2
x 2 3 x 2 log x
log
5
x 3 x 2 log x dx x 3 log x c
2
2
31. 2
vi 2x
x 2 1dx vii Differentiate x 3 log x and hence
1
integrate x 2 log x
logx 11
2
x log x x log x 3x 2
3 1
2
d 3
dx x
log 5 log 2
x 2 3 x 2 log x
log
5
x 3 x 2 log x dx x 3 log x c
2
2
3 x 2 log xdx x 3 log x x 2 dx c
32. 2
vi 2x
x 2 1dx vii Differentiate x 3 log x and hence
1
integrate x 2 log x
logx 11
2
x log x x log x 3x 2
3 1
2
d 3
dx x
log 5 log 2
x 2 3 x 2 log x
log
5
x 3 x 2 log x dx x 3 log x c
2
2
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. 2
vi 2x
x 2 1dx vii Differentiate x 3 log x and hence
1
integrate x 2 log x
logx 11
2
x log x x log x 3x 2
3 1
2
d 3
dx x
log 5 log 2
x 2 3 x 2 log x
log
5
x 3 x 2 log x dx x 3 log x c
2
2
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*