4. Integrating Trig
1
sin ax b c
cosax b dx a
sinax b dx
5. Integrating Trig
1
sin ax b c
cosax b dx a
1
sinax b dx a cosax b c
6. Integrating Trig
1
sin ax b c
cosax b dx a
1
sinax b dx a cosax b c
sec 2 ax b dx
7. Integrating Trig
1
cosax b dx a sin ax b c
1
sinax b dx a cosax b c
1
sec ax b dx a tanax b c
2
8. Integrating Trig
1
cosax b dx a sin ax b c
1
sinax b dx a cosax b c
1
sec ax b dx a tanax b c
2
e.g. i sin 3 xdx
9. Integrating Trig
1
cosax b dx a sin ax b c
1
sinax b dx a cosax b c
1
sec ax b dx a tanax b c
2
1
e.g. i sin 3 xdx cos 3 x c
3
10. Integrating Trig
1
cosax b dx a sin ax b c
1
sinax b dx a cosax b c
1
sec ax b dx a tanax b c
2
1
e.g. i sin 3 xdx cos 3 x c
3
ii cos1 5 x dx
11. Integrating Trig
1
cosax b dx a sin ax b c
1
sinax b dx a cosax b c
1
sec ax b dx a tanax b c
2
1
e.g. i sin 3 xdx cos 3 x c
3
ii cos1 5 x dx 1 sin 1 5 x c
5
12. Integrating Trig
1
cosax b dx a sin ax b c
1
sinax b dx a cosax b c
1
sec ax b dx a tanax b c
2
1
e.g. i sin 3 xdx cos 3 x c
3
ii cos1 5 x dx 1 sin 1 5 x c
5
iii sec 2 x dx
2
13. Integrating Trig
1
cosax b dx a sin ax b c
1
sinax b dx a cosax b c
1
sec ax b dx a tanax b c
2
1
e.g. i sin 3 xdx cos 3 x c
3
ii cos1 5 x dx 1 sin 1 5 x c
5
x dx 2 tan x c
iii sec
2
2 2
18.
1 2 v Find the volume of the solid formed
2
iv sin 2 xdx cos 2 x
2 when y sin x between x 0
6
6 1
cos cos and x 1 is rotated around the x axis.
2 3
1 1
1
2 2
3
4
19.
1 2 v Find the volume of the solid formed
2
iv sin 2 xdx cos 2 x
2 when y sin x between x 0
6
6 1
cos cos and x 1 is rotated around the x axis.
2 3
1 1 V y 2 dx
1
2 2 1
3 sin xdx
0
4
20.
1 2 v Find the volume of the solid formed
2
iv sin 2 xdx cos 2 x
2 when y sin x between x 0
6
6 1
cos cos and x 1 is rotated around the x axis.
2 3
1 1 V y 2 dx
1
2 2 1
3 sin xdx
0
4 1
cos x
1
0
21.
1 2 v Find the volume of the solid formed
2
iv sin 2 xdx cos 2 x
2 when y sin x between x 0
6
6 1
cos cos and x 1 is rotated around the x axis.
2 3
1 1 V y 2 dx
1
2 2 1
3 sin xdx
0
4 1
cos x
1
0
cos cos 0
1 1
2 units 3
22.
1 2 v Find the volume of the solid formed
2
iv sin 2 xdx cos 2 x
2 when y sin x between x 0
6
6 1
cos cos and x 1 is rotated around the x axis.
2 3
1 1 V y 2 dx
1
2 2 1
3 sin xdx
0
4 1
cos x
1
0
cos cos 0
1 1
2 units 3
vi x sec 2 x 2 dx
23.
1 2 v Find the volume of the solid formed
2
iv sin 2 xdx cos 2 x
2 when y sin x between x 0
6
6 1
cos cos and x 1 is rotated around the x axis.
2 3
1 1 V y 2 dx
1
2 2 1
3 sin xdx
0
4 1
cos x
1
0
cos cos 0
1 1
1 2 units 3
vi x sec x dx
2
2 2
2 x sec 2 x 2 dx
24.
1 2 v Find the volume of the solid formed
2
iv sin 2 xdx cos 2 x
2 when y sin x between x 0
6
6 1
cos cos and x 1 is rotated around the x axis.
2 3
1 1 V y 2 dx
1
2 2 1
3 sin xdx
0
4 1
cos x
1
0
cos cos 0
1 1
1 2 units 3
vi x sec x dx
2
2 2
2 x sec 2 x 2 dx
1
tan x 2 c
2
28. vii sin xdx
2
cos2 cos 2 sin 2
1
1 2 sin 2 sin 2 1 cos 2
2
29. vii sin xdx
2
cos2 cos 2 sin 2
1
1 2 sin 2 sin 2 1 cos 2
2
1
2 cos 2 1 cos 2 1 cos 2
2
30. vii sin xdx
2
cos2 cos 2 sin 2
1
1 1 2 sin 2 sin 2 1 cos 2
2 1 cos 2 x dx 2
1
2 cos 2 1 cos 2 1 cos 2
2
31. vii sin xdx
2
cos2 cos 2 sin 2
1
1 1 2 sin 2 sin 2 1 cos 2
2 1 cos 2 x dx 2
1
2 cos 2 1 cos 2 1 cos 2
x sin 2 x c
1 1 2
2 2
x 1
sin 2 x c
2 4
32. vii sin xdx
2
cos2 cos 2 sin 2
1
1 1 2 sin 2 sin 2 1 cos 2
2 1 cos 2 x dx 2
1
2 cos 2 1 cos 2 1 cos 2
x sin 2 x c
1 1 2
2 2
x 1
sin 2 x c
2 4
vii tan xdx
33. vii sin xdx
2
cos2 cos 2 sin 2
1
1 1 2 sin 2 sin 2 1 cos 2
2 1 cos 2 x dx 2
1
2 cos 2 1 cos 2 1 cos 2
x sin 2 x c
1 1 2
2 2
x 1
sin 2 x c
2 4
vii tan xdx sin x dx
cos x
34. vii sin xdx
2
cos2 cos 2 sin 2
1
1 1 2 sin 2 sin 2 1 cos 2
2 1 cos 2 x dx 2
1
2 cos 2 1 cos 2 1 cos 2
x sin 2 x c
1 1 2
2 2
x 1
sin 2 x c
2 4
vii tan xdx sin x dx
cos x
sin x
dx
cos x
35. vii sin xdx
2
cos2 cos 2 sin 2
1
1 1 2 sin 2 sin 2 1 cos 2
2 1 cos 2 x dx 2
1
2 cos 2 1 cos 2 1 cos 2
x sin 2 x c
1 1 2
2 2
x 1
sin 2 x c
2 4
vii tan xdx sin x dx
cos x
sin x
dx
cos x
log cos x c
36. vii sin xdx
2
cos2 cos 2 sin 2
1
1 1 2 sin 2 sin 2 1 cos 2
2 1 cos 2 x dx 2
1
2 cos 2 1 cos 2 1 cos 2
x sin 2 x c
1 1 2
2 2
x 1
sin 2 x c
2 4
vii tan xdx sin x dx
cos x
sin x
dx
cos x
log cos x c
logcos x c
1
log sec x c