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X2 t04 02 trig integrals (2012)

N
Nigel Simmons
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Trig Integrals
Trig Integrals (1) Standard Integrals 1  sin axdx   a cos ax  c
Trig Integrals (1) Standard Integrals 1  sin axdx   a cos ax  c 1  cos axdx  a sin ax  c
Trig Integrals (1) Standard Integrals 1  sin axdx   a cos ax  c 1  cos axdx  a sin ax  c 1  sec 2 axdx  tan ax  c a
Trig Integrals (1) Standard Integrals 1  sin axdx   a cos ax  c 1  cos axdx  a sin ax  c 1  sec 2 axdx  tan ax  c a sin ax  tan axdx   cos ax dx
Trig Integrals (1) Standard Integrals 1  sin axdx   a cos ax  c 1  cos axdx  a sin ax  c 1  sec 2 axdx  tan ax  c a sin ax  tan axdx   cos ax dx 1 1   log cos ax  c OR log sec ax  c a a
2 sin n x or cos n x
2 sin n x or cos n x  sin xdx   cos x  c
2 sin n x or cos n x  sin xdx   cos x  c  sin 2 xdx
2 sin n x or cos n x  sin xdx   cos x  c 1  sin xdx  2  1  cos 2 x dx 2
2 sin n x or cos n x  sin xdx   cos x  c 1  sin xdx  2  1  cos 2 x dx 2 1 1    x  sin 2 x   c 2 2 
2 sin n x or cos n x  sin xdx   cos x  c 1  sin xdx  2  1  cos 2 x dx 2 1 1    x  sin 2 x   c 2 2   sin 3 xdx
2 sin n x or cos n x  sin xdx   cos x  c 1  sin xdx  2  1  cos 2 x dx 2 1 1    x  sin 2 x   c 2 2   sin 3 xdx   sin x sin xdx 2
2 sin n x or cos n x  sin xdx   cos x  c 1  sin xdx  2  1  cos 2 x dx 2 1 1    x  sin 2 x   c 2 2   sin 3 xdx   sin x sin xdx 2   sin x1  cos 2 x dx
2 sin n x or cos n x  sin xdx   cos x  c 1  sin xdx  2  1  cos 2 x dx 2 1 1    x  sin 2 x   c 2 2   sin 3 xdx   sin x sin xdx 2   sin x1  cos 2 x dx u  cos x du   sin xdx
2 sin n x or cos n x  sin xdx   cos x  c 1  sin xdx  2  1  cos 2 x dx 2 1 1    x  sin 2 x   c 2 2   sin 3 xdx   sin x sin xdx 2   sin x1  cos 2 x dx u  cos x du   sin xdx    1  u 2 du
2 sin n x or cos n x  sin xdx   cos x  c 1  sin xdx  2  1  cos 2 x dx 2 1 1    x  sin 2 x   c 2 2   sin 3 xdx   sin x sin xdx 2   sin x1  cos 2 x dx u  cos x du   sin xdx    1  u 2 du 1 3  u u c 3 1 3  cos x  cos x  c 3
2 sin n x or cos n x  sin xdx   cos x  c 1  sin xdx  2  1  cos 2 x dx 2 1 1    x  sin 2 x   c 2 2   sin 3 xdx   sin x sin xdx 2   sin x1  cos 2 x dx u  cos x du   sin xdx    1  u 2 du Odd Power 1 3  u u c Factorise as sin xsin x  2 some power 3 1 3 Substitute sin 2 x  1  cos 2 x  cos x  cos x  c 3 Use u  cos x
 sin 4 xdx
 sin xdx   sin x  dx 4 2 2
 sin xdx   sin x  dx 4 2 2 1   1  cos 2 x  dx 2 4
 sin xdx   sin x  dx 4 2 2 1   1  cos 2 x  dx 2 4   1  2 cos 2 x  cos 2 2 x dx 1 4
 sin xdx   sin x  dx 4 2 2 1   1  cos 2 x  dx 2 4   1  2 cos 2 x  cos 2 2 x dx 1 4   1  2 cos 2 x  1  cos 4 x  dx 1  1 4  2  
 sin xdx   sin x  dx 4 2 2 1   1  cos 2 x  dx 2 4   1  2 cos 2 x  cos 2 2 x dx 1 4   1  2 cos 2 x  1  cos 4 x  dx 1  1 4  2       2 cos 2 x  cos 4 x dx 1 3 1  4 2 2 
 sin xdx   sin x  dx 4 2 2 1   1  cos 2 x  dx 2 4   1  2 cos 2 x  cos 2 2 x dx 1 4   1  2 cos 2 x  1  cos 4 x  dx 1  1 4  2       2 cos 2 x  cos 4 x dx 1 3 1  4 2 2    x  sin 2 x  sin 4 x   c 13 1  42 8 
xdx   sin x  dx Even Power  sin 4 2 2 Factorise as sin x  2 some power 1   1  cos 2 x  dx 2 1 4 Substitute sin x  1  cos 2 x  2 2   1  2 cos 2 x  cos 2 2 x dx 1 4   1  2 cos 2 x  1  cos 4 x  dx 1  1 4  2       2 cos 2 x  cos 4 x dx 1 3 1  4 2 2    x  sin 2 x  sin 4 x   c 13 1  42 8 
 sin 5 xdx
 sin xdx   sin xsin x  dx 5 2 2
 sin xdx   sin xsin x  dx 5 2 2   sin x1  cos x  dx 2 2
 sin xdx   sin xsin x  dx 5 2 2   sin x1  cos x  dx u  cos x 2 2 du   sin xdx
 sin xdx   sin xsin x  dx 5 2 2   sin x1  cos x  dx u  cos x 2 2 du   sin xdx    1  u  du 2 2    1  2u 2  u 4 du
 sin xdx   sin xsin x  dx 5 2 2   sin x1  cos x  dx u  cos x 2 2 du   sin xdx    1  u  du 2 2    1  2u 2  u 4 du  u  2 u3  1 u5   c     3 5  2 1   cos x  cos3 x  cos 5 x  c 3 5
3 sin n x and cos n x
3 sin n x and cos n x Usually done by substitution u  sin x or u  cos x
3 sin n x and cos n x Usually done by substitution u  sin x or u  cos x e.g. i  cos 5 x sin 3 xdx
3 sin n x and cos n x Usually done by substitution u  sin x or u  cos x e.g. i  cos 5 x sin 3 xdx   cos5 x1  cos 2 x sin xdx
3 sin n x and cos n x Usually done by substitution u  sin x or u  cos x e.g. i  cos 5 x sin 3 xdx   cos5 x1  cos 2 x sin xdx u  cos x du   sin xdx
3 sin n x and cos n x Usually done by substitution u  sin x or u  cos x e.g. i  cos 5 x sin 3 xdx   cos5 x1  cos 2 x sin xdx u  cos x    u 5 1  u 2 du du   sin xdx   u 7  u 5 du
3 sin n x and cos n x Usually done by substitution u  sin x or u  cos x e.g. i  cos 5 x sin 3 xdx   cos5 x1  cos 2 x sin xdx u  cos x    u 5 1  u 2 du du   sin xdx   u 7  u 5 du 1 8 1 6  u  u c 8 6 1 8 1  cos x  cos 6 x  c 8 6
3 sin n x and cos n x Usually done by substitution u  sin x or u  cos x e.g. i  cos 5 x sin 3 xdx   cos5 x1  cos 2 x sin xdx u  cos x    u 5 1  u 2 du du   sin xdx   u 7  u 5 du Both powers odd 1 8 1 6  u  u c Choose either as u 8 6 Usually the higher power 1 8 1  cos x  cos 6 x  c 8 6
ii  sin 6 x cos3 xdx
ii  sin 6 x cos3 xdx   sin 6 x1  sin 2 x cos xdx
ii  sin 6 x cos3 xdx   sin 6 x1  sin 2 x cos xdx u  sin x du  cos xdx
ii  sin 6 x cos3 xdx   sin 6 x1  sin 2 x cos xdx u  sin x   u 6 1  u 2 du du  cos xdx   u 6  u 8 du
ii  sin 6 x cos3 xdx   sin 6 x1  sin 2 x cos xdx u  sin x   u 6 1  u 2 du du  cos xdx   u 6  u 8 du 1 7 1 9  u  u c 7 9 1 7 1 9  sin x  sin x  c 7 9
ii  sin 6 x cos3 xdx   sin 6 x1  sin 2 x cos xdx u  sin x   u 6 1  u 2 du du  cos xdx   u 6  u 8 du One power odd & one power even 1 7 1 9  u  u c Choose even as u 7 9 1 7 1 9  sin x  sin x  c 7 9
ii  sin 6 x cos3 xdx   sin 6 x1  sin 2 x cos xdx u  sin x   u 6 1  u 2 du du  cos xdx   u 6  u 8 du One power odd & one power even 1 7 1 9  u  u c Choose even as u 7 9 1 7 1 9  sin x  sin x  c 7 9 iii  sin 2 x cos 2 xdx
ii  sin 6 x cos3 xdx   sin 6 x1  sin 2 x cos xdx u  sin x   u 6 1  u 2 du du  cos xdx   u 6  u 8 du One power odd & one power even 1 7 1 9  u  u c Choose even as u 7 9 1 7 1 9  sin x  sin x  c 7 9 iii  sin 2 x cos 2 xdx   sin 2 x1  sin 2 x dx
ii  sin 6 x cos3 xdx   sin 6 x1  sin 2 x cos xdx u  sin x   u 6 1  u 2 du du  cos xdx   u 6  u 8 du One power odd & one power even 1 7 1 9  u  u c Choose even as u 7 9 1 7 1 9  sin x  sin x  c 7 9 iii  sin 2 x cos 2 xdx   sin 2 x1  sin 2 x dx   sin 2 x  sin 4 x dx
ii  sin 6 x cos3 xdx   sin 6 x1  sin 2 x cos xdx u  sin x   u 6 1  u 2 du du  cos xdx   u 6  u 8 du One power odd & one power even 1 7 1 9  u  u c Choose even as u 7 9 1 7 1 9  sin x  sin x  c 7 9 iii  sin 2 x cos 2 xdx   sin 2 x1  sin 2 x dx   sin 2 x  sin 4 x dx 1 1 3 1 1  x  sin 2 x  x  sin 2 x  sin 4 x  c 2 4 8 4 32
ii  sin 6 x cos3 xdx   sin 6 x1  sin 2 x cos xdx u  sin x   u 6 1  u 2 du du  cos xdx   u 6  u 8 du One power odd & one power even 1 7 1 9  u  u c Choose even as u 7 9 1 7 1 9  sin x  sin x  c 7 9 iii  sin 2 x cos 2 xdx   sin 2 x1  sin 2 x dx   sin 2 x  sin 4 x dx 1 1 3 1 1  x  sin 2 x  x  sin 2 x  sin 4 x  c 2 4 8 4 32 1 1  x  sin 4 x  c 8 32
ii  sin 6 x cos3 xdx   sin 6 x1  sin 2 x cos xdx u  sin x   u 6 1  u 2 du du  cos xdx   u 6  u 8 du One power odd & one power even 1 7 1 9  u  u c Choose even as u 7 9 1 7 1 9  sin x  sin x  c 7 9 iii  sin 2 x cos 2 xdx   sin 2 x1  sin 2 x dx   sin 2 x  sin 4 x dx Both powers even 1 1 3 1 1 Use sin x  1  cos x 2 2  x  sin 2 x  x  sin 2 x  sin 4 x  c 2 4 8 4 32 or cos 2 x  1  sin 2 x 1 1  x  sin 4 x  c 8 32
4 tan n x or cot n x
4 tan n x or cot n x  tan xdx   log cos x  c
4 tan n x or cot n x  tan xdx   log cos x  c  tan xdx 2
4 tan n x or cot n x  tan xdx   log cos x  c  tan xdx   sec x  1dx 2 2
4 tan n x or cot n x  tan xdx   log cos x  c  tan xdx   sec x  1dx 2 2  tan x  x  c
4 tan n x or cot n x  tan xdx   log cos x  c  tan xdx   sec x  1dx 2 2  tan x  x  c  tan 3 xdx
4 tan n x or cot n x  tan xdx   log cos x  c  tan xdx   sec x  1dx 2 2  tan x  x  c  tan 3 xdx   tan xsec 2 x  1dx   tan x sec 2 xdx   tan xdx
4 tan n x or cot n x  tan xdx   log cos x  c  tan xdx   sec x  1dx 2 2  tan x  x  c  tan 3 xdx   tan xsec 2 x  1dx   tan x sec 2 xdx   tan xdx u  tan x du  sec 2 xdx
4 tan n x or cot n x  tan xdx   log cos x  c  tan xdx   sec x  1dx 2 2  tan x  x  c  tan 3 xdx   tan xsec 2 x  1dx   tan x sec 2 xdx   tan xdx u  tan x   udu   tan xdx du  sec 2 xdx
4 tan n x or cot n x  tan xdx   log cos x  c  tan xdx   sec x  1dx 2 2  tan x  x  c  tan 3 xdx   tan xsec 2 x  1dx   tan x sec 2 xdx   tan xdx u  tan x   udu   tan xdx du  sec 2 xdx 1  u 2  log cos x  c 2 1 2  tan x  log cos x  c 2
 tan 4 xdx
 tan 4 xdx   tan 2 xsec 2 x  1dx   tan 2 x sec 2 xdx   tan 2 xdx
 tan 4 xdx   tan 2 xsec 2 x  1dx   tan 2 x sec 2 xdx   tan 2 xdx u  tan x du  sec 2 xdx
 tan 4 xdx   tan 2 xsec 2 x  1dx   tan 2 x sec 2 xdx   tan 2 xdx u  tan x   u du   tan xdx 2 2 du  sec 2 xdx
 tan 4 xdx   tan 2 xsec 2 x  1dx   tan 2 x sec 2 xdx   tan 2 xdx u  tan x   u du   tan xdx 2 2 du  sec 2 xdx 1 3  u  tan x  x  c 3 1 3  tan x  tan x  x  c 3
5 sec n x or cosecn x
5 sec n x or cosecn x  sec xdx
5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx
5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x
5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c
5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c  sec 2 xdx  tan x  c
5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c  sec 2 xdx  tan x  c  sec3 xdx
5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c  sec 2 xdx  tan x  c  sec3 xdx   sec x sec 2 xdx
5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c  sec 2 xdx  tan x  c  sec3 xdx   sec x sec 2 xdx u  sec x
5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c  sec 2 xdx  tan x  c  sec3 xdx   sec x sec 2 xdx u  sec x du  sec x tan xdx
5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c  sec 2 xdx  tan x  c  sec3 xdx   sec x sec 2 xdx u  sec x du  sec x tan xdx dv  sec 2 xdx
5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c  sec 2 xdx  tan x  c  sec3 xdx   sec x sec 2 xdx u  sec x v  tan x du  sec x tan xdx dv  sec 2 xdx
5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c  sec 2 xdx  tan x  c  sec3 xdx   sec x sec 2 xdx u  sec x v  tan x  sec x tan x   sec x tan 2 xdx du  sec x tan xdx dv  sec 2 xdx
5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c  sec 2 xdx  tan x  c  sec3 xdx   sec x sec 2 xdx u  sec x v  tan x  sec x tan x   sec x tan 2 xdx du  sec x tan xdx dv  sec 2 xdx  sec x tan x   sec xsec 2 x  1dx  sec x tan x   sec3 xdx   sec xdx  sec x tan x   sec3 xdx  logsec x  tan x 
5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c  sec 2 xdx  tan x  c  sec3 xdx   sec x sec 2 xdx u  sec x v  tan x  sec x tan x   sec x tan 2 xdx du  sec x tan xdx dv  sec 2 xdx  sec x tan x   sec xsec 2 x  1dx  sec x tan x   sec3 xdx   sec xdx  sec x tan x   sec3 xdx  logsec x  tan x   2  sec3 xdx  sec x tan x  logsec x  tan x   c
5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c  sec 2 xdx  tan x  c  sec3 xdx   sec x sec 2 xdx u  sec x v  tan x  sec x tan x   sec x tan 2 xdx du  sec x tan xdx dv  sec 2 xdx  sec x tan x   sec xsec 2 x  1dx  sec x tan x   sec3 xdx   sec xdx  sec x tan x   sec3 xdx  logsec x  tan x   2  sec3 xdx  sec x tan x  logsec x  tan x   c 1 1  sec3 xdx  sec x tan x  logsec x  tan x   c 2 2
5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c  sec 2 xdx  tan x  c  sec3 xdx   sec x sec 2 xdx u  sec x v  tan x  sec x tan x   sec x tan 2 xdx du  sec x tan xdx dv  sec 2 xdx  sec x tan x   sec xsec 2 x  1dx Odd powers  sec x tan x   sec xdx   sec xdx 3 Done by parts  sec x tan x   sec3 xdx  logsec x  tan x   2  sec3 xdx  sec x tan x  logsec x  tan x   c 1 1  sec3 xdx  sec x tan x  logsec x  tan x   c 2 2
 sec 4 xdx
 sec 4 xdx   sec 2 x1  tan 2 x dx
 sec 4 xdx   sec 2 x1  tan 2 x dx u  tan x du  sec 2 xdx
 sec 4 xdx   sec 2 x1  tan 2 x dx u  tan x   1  u du 2 du  sec 2 xdx
 sec 4 xdx   sec 2 x1  tan 2 x dx u  tan x   1  u du 2 du  sec 2 xdx 1 3 u u c 3 1 3  tan x  tan x  c 3
 sec 4 xdx   sec 2 x1  tan 2 x dx u  tan x   1  u du 2 du  sec 2 xdx 1 3 Even Power u u c Factorise as sec xsec x  3 2 2 some power 1 3  tan x  tan x  c Substitute sec 2 x  1  tan 2 x 3 Use u  tan x
 sec 4 xdx   sec 2 x1  tan 2 x dx u  tan x   1  u du 2 du  sec 2 xdx 1 3 Even Power u u c Factorise as sec xsec x  3 2 2 some power 1 3  tan x  tan x  c Substitute sec 2 x  1  tan 2 x 3 Use u  tan x Exercise 2C; 1, 2, 4, 5, 7, 8, 10, 12, 14, 16, 17, 18

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X2 t04 02 trig integrals (2012)

  • 2. Trig Integrals (1) Standard Integrals 1  sin axdx   a cos ax  c
  • 3. Trig Integrals (1) Standard Integrals 1  sin axdx   a cos ax  c 1  cos axdx  a sin ax  c
  • 4. Trig Integrals (1) Standard Integrals 1  sin axdx   a cos ax  c 1  cos axdx  a sin ax  c 1  sec 2 axdx  tan ax  c a
  • 5. Trig Integrals (1) Standard Integrals 1  sin axdx   a cos ax  c 1  cos axdx  a sin ax  c 1  sec 2 axdx  tan ax  c a sin ax  tan axdx   cos ax dx
  • 6. Trig Integrals (1) Standard Integrals 1  sin axdx   a cos ax  c 1  cos axdx  a sin ax  c 1  sec 2 axdx  tan ax  c a sin ax  tan axdx   cos ax dx 1 1   log cos ax  c OR log sec ax  c a a
  • 7. 2 sin n x or cos n x
  • 8. 2 sin n x or cos n x  sin xdx   cos x  c
  • 9. 2 sin n x or cos n x  sin xdx   cos x  c  sin 2 xdx
  • 10. 2 sin n x or cos n x  sin xdx   cos x  c 1  sin xdx  2  1  cos 2 x dx 2
  • 11. 2 sin n x or cos n x  sin xdx   cos x  c 1  sin xdx  2  1  cos 2 x dx 2 1 1    x  sin 2 x   c 2 2 
  • 12. 2 sin n x or cos n x  sin xdx   cos x  c 1  sin xdx  2  1  cos 2 x dx 2 1 1    x  sin 2 x   c 2 2   sin 3 xdx
  • 13. 2 sin n x or cos n x  sin xdx   cos x  c 1  sin xdx  2  1  cos 2 x dx 2 1 1    x  sin 2 x   c 2 2   sin 3 xdx   sin x sin xdx 2
  • 14. 2 sin n x or cos n x  sin xdx   cos x  c 1  sin xdx  2  1  cos 2 x dx 2 1 1    x  sin 2 x   c 2 2   sin 3 xdx   sin x sin xdx 2   sin x1  cos 2 x dx
  • 15. 2 sin n x or cos n x  sin xdx   cos x  c 1  sin xdx  2  1  cos 2 x dx 2 1 1    x  sin 2 x   c 2 2   sin 3 xdx   sin x sin xdx 2   sin x1  cos 2 x dx u  cos x du   sin xdx
  • 16. 2 sin n x or cos n x  sin xdx   cos x  c 1  sin xdx  2  1  cos 2 x dx 2 1 1    x  sin 2 x   c 2 2   sin 3 xdx   sin x sin xdx 2   sin x1  cos 2 x dx u  cos x du   sin xdx    1  u 2 du
  • 17. 2 sin n x or cos n x  sin xdx   cos x  c 1  sin xdx  2  1  cos 2 x dx 2 1 1    x  sin 2 x   c 2 2   sin 3 xdx   sin x sin xdx 2   sin x1  cos 2 x dx u  cos x du   sin xdx    1  u 2 du 1 3  u u c 3 1 3  cos x  cos x  c 3
  • 18. 2 sin n x or cos n x  sin xdx   cos x  c 1  sin xdx  2  1  cos 2 x dx 2 1 1    x  sin 2 x   c 2 2   sin 3 xdx   sin x sin xdx 2   sin x1  cos 2 x dx u  cos x du   sin xdx    1  u 2 du Odd Power 1 3  u u c Factorise as sin xsin x  2 some power 3 1 3 Substitute sin 2 x  1  cos 2 x  cos x  cos x  c 3 Use u  cos x
  • 19.  sin 4 xdx
  • 20.  sin xdx   sin x  dx 4 2 2
  • 21.  sin xdx   sin x  dx 4 2 2 1   1  cos 2 x  dx 2 4
  • 22.  sin xdx   sin x  dx 4 2 2 1   1  cos 2 x  dx 2 4   1  2 cos 2 x  cos 2 2 x dx 1 4
  • 23.  sin xdx   sin x  dx 4 2 2 1   1  cos 2 x  dx 2 4   1  2 cos 2 x  cos 2 2 x dx 1 4   1  2 cos 2 x  1  cos 4 x  dx 1  1 4  2  
  • 24.  sin xdx   sin x  dx 4 2 2 1   1  cos 2 x  dx 2 4   1  2 cos 2 x  cos 2 2 x dx 1 4   1  2 cos 2 x  1  cos 4 x  dx 1  1 4  2       2 cos 2 x  cos 4 x dx 1 3 1  4 2 2 
  • 25.  sin xdx   sin x  dx 4 2 2 1   1  cos 2 x  dx 2 4   1  2 cos 2 x  cos 2 2 x dx 1 4   1  2 cos 2 x  1  cos 4 x  dx 1  1 4  2       2 cos 2 x  cos 4 x dx 1 3 1  4 2 2    x  sin 2 x  sin 4 x   c 13 1  42 8 
  • 26. xdx   sin x  dx Even Power  sin 4 2 2 Factorise as sin x  2 some power 1   1  cos 2 x  dx 2 1 4 Substitute sin x  1  cos 2 x  2 2   1  2 cos 2 x  cos 2 2 x dx 1 4   1  2 cos 2 x  1  cos 4 x  dx 1  1 4  2       2 cos 2 x  cos 4 x dx 1 3 1  4 2 2    x  sin 2 x  sin 4 x   c 13 1  42 8 
  • 27.  sin 5 xdx
  • 28.  sin xdx   sin xsin x  dx 5 2 2
  • 29.  sin xdx   sin xsin x  dx 5 2 2   sin x1  cos x  dx 2 2
  • 30.  sin xdx   sin xsin x  dx 5 2 2   sin x1  cos x  dx u  cos x 2 2 du   sin xdx
  • 31.  sin xdx   sin xsin x  dx 5 2 2   sin x1  cos x  dx u  cos x 2 2 du   sin xdx    1  u  du 2 2    1  2u 2  u 4 du
  • 32.  sin xdx   sin xsin x  dx 5 2 2   sin x1  cos x  dx u  cos x 2 2 du   sin xdx    1  u  du 2 2    1  2u 2  u 4 du  u  2 u3  1 u5   c     3 5  2 1   cos x  cos3 x  cos 5 x  c 3 5
  • 33. 3 sin n x and cos n x
  • 34. 3 sin n x and cos n x Usually done by substitution u  sin x or u  cos x
  • 35. 3 sin n x and cos n x Usually done by substitution u  sin x or u  cos x e.g. i  cos 5 x sin 3 xdx
  • 36. 3 sin n x and cos n x Usually done by substitution u  sin x or u  cos x e.g. i  cos 5 x sin 3 xdx   cos5 x1  cos 2 x sin xdx
  • 37. 3 sin n x and cos n x Usually done by substitution u  sin x or u  cos x e.g. i  cos 5 x sin 3 xdx   cos5 x1  cos 2 x sin xdx u  cos x du   sin xdx
  • 38. 3 sin n x and cos n x Usually done by substitution u  sin x or u  cos x e.g. i  cos 5 x sin 3 xdx   cos5 x1  cos 2 x sin xdx u  cos x    u 5 1  u 2 du du   sin xdx   u 7  u 5 du
  • 39. 3 sin n x and cos n x Usually done by substitution u  sin x or u  cos x e.g. i  cos 5 x sin 3 xdx   cos5 x1  cos 2 x sin xdx u  cos x    u 5 1  u 2 du du   sin xdx   u 7  u 5 du 1 8 1 6  u  u c 8 6 1 8 1  cos x  cos 6 x  c 8 6
  • 40. 3 sin n x and cos n x Usually done by substitution u  sin x or u  cos x e.g. i  cos 5 x sin 3 xdx   cos5 x1  cos 2 x sin xdx u  cos x    u 5 1  u 2 du du   sin xdx   u 7  u 5 du Both powers odd 1 8 1 6  u  u c Choose either as u 8 6 Usually the higher power 1 8 1  cos x  cos 6 x  c 8 6
  • 41. ii  sin 6 x cos3 xdx
  • 42. ii  sin 6 x cos3 xdx   sin 6 x1  sin 2 x cos xdx
  • 43. ii  sin 6 x cos3 xdx   sin 6 x1  sin 2 x cos xdx u  sin x du  cos xdx
  • 44. ii  sin 6 x cos3 xdx   sin 6 x1  sin 2 x cos xdx u  sin x   u 6 1  u 2 du du  cos xdx   u 6  u 8 du
  • 45. ii  sin 6 x cos3 xdx   sin 6 x1  sin 2 x cos xdx u  sin x   u 6 1  u 2 du du  cos xdx   u 6  u 8 du 1 7 1 9  u  u c 7 9 1 7 1 9  sin x  sin x  c 7 9
  • 46. ii  sin 6 x cos3 xdx   sin 6 x1  sin 2 x cos xdx u  sin x   u 6 1  u 2 du du  cos xdx   u 6  u 8 du One power odd & one power even 1 7 1 9  u  u c Choose even as u 7 9 1 7 1 9  sin x  sin x  c 7 9
  • 47. ii  sin 6 x cos3 xdx   sin 6 x1  sin 2 x cos xdx u  sin x   u 6 1  u 2 du du  cos xdx   u 6  u 8 du One power odd & one power even 1 7 1 9  u  u c Choose even as u 7 9 1 7 1 9  sin x  sin x  c 7 9 iii  sin 2 x cos 2 xdx
  • 48. ii  sin 6 x cos3 xdx   sin 6 x1  sin 2 x cos xdx u  sin x   u 6 1  u 2 du du  cos xdx   u 6  u 8 du One power odd & one power even 1 7 1 9  u  u c Choose even as u 7 9 1 7 1 9  sin x  sin x  c 7 9 iii  sin 2 x cos 2 xdx   sin 2 x1  sin 2 x dx
  • 49. ii  sin 6 x cos3 xdx   sin 6 x1  sin 2 x cos xdx u  sin x   u 6 1  u 2 du du  cos xdx   u 6  u 8 du One power odd & one power even 1 7 1 9  u  u c Choose even as u 7 9 1 7 1 9  sin x  sin x  c 7 9 iii  sin 2 x cos 2 xdx   sin 2 x1  sin 2 x dx   sin 2 x  sin 4 x dx
  • 50. ii  sin 6 x cos3 xdx   sin 6 x1  sin 2 x cos xdx u  sin x   u 6 1  u 2 du du  cos xdx   u 6  u 8 du One power odd & one power even 1 7 1 9  u  u c Choose even as u 7 9 1 7 1 9  sin x  sin x  c 7 9 iii  sin 2 x cos 2 xdx   sin 2 x1  sin 2 x dx   sin 2 x  sin 4 x dx 1 1 3 1 1  x  sin 2 x  x  sin 2 x  sin 4 x  c 2 4 8 4 32
  • 51. ii  sin 6 x cos3 xdx   sin 6 x1  sin 2 x cos xdx u  sin x   u 6 1  u 2 du du  cos xdx   u 6  u 8 du One power odd & one power even 1 7 1 9  u  u c Choose even as u 7 9 1 7 1 9  sin x  sin x  c 7 9 iii  sin 2 x cos 2 xdx   sin 2 x1  sin 2 x dx   sin 2 x  sin 4 x dx 1 1 3 1 1  x  sin 2 x  x  sin 2 x  sin 4 x  c 2 4 8 4 32 1 1  x  sin 4 x  c 8 32
  • 52. ii  sin 6 x cos3 xdx   sin 6 x1  sin 2 x cos xdx u  sin x   u 6 1  u 2 du du  cos xdx   u 6  u 8 du One power odd & one power even 1 7 1 9  u  u c Choose even as u 7 9 1 7 1 9  sin x  sin x  c 7 9 iii  sin 2 x cos 2 xdx   sin 2 x1  sin 2 x dx   sin 2 x  sin 4 x dx Both powers even 1 1 3 1 1 Use sin x  1  cos x 2 2  x  sin 2 x  x  sin 2 x  sin 4 x  c 2 4 8 4 32 or cos 2 x  1  sin 2 x 1 1  x  sin 4 x  c 8 32
  • 53. 4 tan n x or cot n x
  • 54. 4 tan n x or cot n x  tan xdx   log cos x  c
  • 55. 4 tan n x or cot n x  tan xdx   log cos x  c  tan xdx 2
  • 56. 4 tan n x or cot n x  tan xdx   log cos x  c  tan xdx   sec x  1dx 2 2
  • 57. 4 tan n x or cot n x  tan xdx   log cos x  c  tan xdx   sec x  1dx 2 2  tan x  x  c
  • 58. 4 tan n x or cot n x  tan xdx   log cos x  c  tan xdx   sec x  1dx 2 2  tan x  x  c  tan 3 xdx
  • 59. 4 tan n x or cot n x  tan xdx   log cos x  c  tan xdx   sec x  1dx 2 2  tan x  x  c  tan 3 xdx   tan xsec 2 x  1dx   tan x sec 2 xdx   tan xdx
  • 60. 4 tan n x or cot n x  tan xdx   log cos x  c  tan xdx   sec x  1dx 2 2  tan x  x  c  tan 3 xdx   tan xsec 2 x  1dx   tan x sec 2 xdx   tan xdx u  tan x du  sec 2 xdx
  • 61. 4 tan n x or cot n x  tan xdx   log cos x  c  tan xdx   sec x  1dx 2 2  tan x  x  c  tan 3 xdx   tan xsec 2 x  1dx   tan x sec 2 xdx   tan xdx u  tan x   udu   tan xdx du  sec 2 xdx
  • 62. 4 tan n x or cot n x  tan xdx   log cos x  c  tan xdx   sec x  1dx 2 2  tan x  x  c  tan 3 xdx   tan xsec 2 x  1dx   tan x sec 2 xdx   tan xdx u  tan x   udu   tan xdx du  sec 2 xdx 1  u 2  log cos x  c 2 1 2  tan x  log cos x  c 2
  • 63.  tan 4 xdx
  • 64.  tan 4 xdx   tan 2 xsec 2 x  1dx   tan 2 x sec 2 xdx   tan 2 xdx
  • 65.  tan 4 xdx   tan 2 xsec 2 x  1dx   tan 2 x sec 2 xdx   tan 2 xdx u  tan x du  sec 2 xdx
  • 66.  tan 4 xdx   tan 2 xsec 2 x  1dx   tan 2 x sec 2 xdx   tan 2 xdx u  tan x   u du   tan xdx 2 2 du  sec 2 xdx
  • 67.  tan 4 xdx   tan 2 xsec 2 x  1dx   tan 2 x sec 2 xdx   tan 2 xdx u  tan x   u du   tan xdx 2 2 du  sec 2 xdx 1 3  u  tan x  x  c 3 1 3  tan x  tan x  x  c 3
  • 68. 5 sec n x or cosecn x
  • 69. 5 sec n x or cosecn x  sec xdx
  • 70. 5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx
  • 71. 5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x
  • 72. 5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c
  • 73. 5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c  sec 2 xdx  tan x  c
  • 74. 5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c  sec 2 xdx  tan x  c  sec3 xdx
  • 75. 5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c  sec 2 xdx  tan x  c  sec3 xdx   sec x sec 2 xdx
  • 76. 5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c  sec 2 xdx  tan x  c  sec3 xdx   sec x sec 2 xdx u  sec x
  • 77. 5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c  sec 2 xdx  tan x  c  sec3 xdx   sec x sec 2 xdx u  sec x du  sec x tan xdx
  • 78. 5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c  sec 2 xdx  tan x  c  sec3 xdx   sec x sec 2 xdx u  sec x du  sec x tan xdx dv  sec 2 xdx
  • 79. 5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c  sec 2 xdx  tan x  c  sec3 xdx   sec x sec 2 xdx u  sec x v  tan x du  sec x tan xdx dv  sec 2 xdx
  • 80. 5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c  sec 2 xdx  tan x  c  sec3 xdx   sec x sec 2 xdx u  sec x v  tan x  sec x tan x   sec x tan 2 xdx du  sec x tan xdx dv  sec 2 xdx
  • 81. 5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c  sec 2 xdx  tan x  c  sec3 xdx   sec x sec 2 xdx u  sec x v  tan x  sec x tan x   sec x tan 2 xdx du  sec x tan xdx dv  sec 2 xdx  sec x tan x   sec xsec 2 x  1dx  sec x tan x   sec3 xdx   sec xdx  sec x tan x   sec3 xdx  logsec x  tan x 
  • 82. 5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c  sec 2 xdx  tan x  c  sec3 xdx   sec x sec 2 xdx u  sec x v  tan x  sec x tan x   sec x tan 2 xdx du  sec x tan xdx dv  sec 2 xdx  sec x tan x   sec xsec 2 x  1dx  sec x tan x   sec3 xdx   sec xdx  sec x tan x   sec3 xdx  logsec x  tan x   2  sec3 xdx  sec x tan x  logsec x  tan x   c
  • 83. 5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c  sec 2 xdx  tan x  c  sec3 xdx   sec x sec 2 xdx u  sec x v  tan x  sec x tan x   sec x tan 2 xdx du  sec x tan xdx dv  sec 2 xdx  sec x tan x   sec xsec 2 x  1dx  sec x tan x   sec3 xdx   sec xdx  sec x tan x   sec3 xdx  logsec x  tan x   2  sec3 xdx  sec x tan x  logsec x  tan x   c 1 1  sec3 xdx  sec x tan x  logsec x  tan x   c 2 2
  • 84. 5 sec n x or cosecn x sec xsec x  tan x   sec xdx  sec x  tan x dx sec 2 x  sec x tan x  dx sec x  tan x  logsec x  tan x   c  sec 2 xdx  tan x  c  sec3 xdx   sec x sec 2 xdx u  sec x v  tan x  sec x tan x   sec x tan 2 xdx du  sec x tan xdx dv  sec 2 xdx  sec x tan x   sec xsec 2 x  1dx Odd powers  sec x tan x   sec xdx   sec xdx 3 Done by parts  sec x tan x   sec3 xdx  logsec x  tan x   2  sec3 xdx  sec x tan x  logsec x  tan x   c 1 1  sec3 xdx  sec x tan x  logsec x  tan x   c 2 2
  • 85.  sec 4 xdx
  • 86.  sec 4 xdx   sec 2 x1  tan 2 x dx
  • 87.  sec 4 xdx   sec 2 x1  tan 2 x dx u  tan x du  sec 2 xdx
  • 88.  sec 4 xdx   sec 2 x1  tan 2 x dx u  tan x   1  u du 2 du  sec 2 xdx
  • 89.  sec 4 xdx   sec 2 x1  tan 2 x dx u  tan x   1  u du 2 du  sec 2 xdx 1 3 u u c 3 1 3  tan x  tan x  c 3
  • 90.  sec 4 xdx   sec 2 x1  tan 2 x dx u  tan x   1  u du 2 du  sec 2 xdx 1 3 Even Power u u c Factorise as sec xsec x  3 2 2 some power 1 3  tan x  tan x  c Substitute sec 2 x  1  tan 2 x 3 Use u  tan x
  • 91.  sec 4 xdx   sec 2 x1  tan 2 x dx u  tan x   1  u du 2 du  sec 2 xdx 1 3 Even Power u u c Factorise as sec xsec x  3 2 2 some power 1 3  tan x  tan x  c Substitute sec 2 x  1  tan 2 x 3 Use u  tan x Exercise 2C; 1, 2, 4, 5, 7, 8, 10, 12, 14, 16, 17, 18