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12X1 T03 04 integrating trig functions

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Nigel Simmons
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Integrating Trig
Integrating Trig  cosax  b dx
Integrating Trig 1  sin ax  b   c  cosax  b dx a
Integrating Trig 1  sin ax  b   c  cosax  b dx a  sinax  b dx
Integrating Trig 1  sin ax  b   c  cosax  b dx a 1  sinax  b dx   a cosax  b   c
Integrating Trig 1  sin ax  b   c  cosax  b dx a 1  sinax  b dx   a cosax  b   c  sec 2 ax  b dx
Integrating Trig 1  cosax  b dx a sin ax  b   c 1  sinax  b dx   a cosax  b   c 1  sec ax  b dx  a tanax  b   c 2
Integrating Trig 1  cosax  b dx a sin ax  b   c 1  sinax  b dx   a cosax  b   c 1  sec ax  b dx  a tanax  b   c 2 e.g. i   sin 3 xdx
Integrating Trig 1  cosax  b dx a  sin ax  b   c 1  sinax  b dx   a cosax  b   c 1  sec ax  b dx  a tanax  b   c 2 1 e.g. i   sin 3 xdx   cos 3 x  c 3
Integrating Trig 1  cosax  b dx a  sin ax  b   c 1  sinax  b dx   a cosax  b   c 1  sec ax  b dx  a tanax  b   c 2 1 e.g. i   sin 3 xdx   cos 3 x  c 3 ii   cos1  5 x dx
Integrating Trig 1  cosax  b dx a  sin ax  b   c 1  sinax  b dx   a cosax  b   c 1  sec ax  b dx  a tanax  b   c 2 1 e.g. i   sin 3 xdx   cos 3 x  c 3 ii   cos1  5 x dx   1 sin 1  5 x   c 5
Integrating Trig 1  cosax  b dx a  sin ax  b   c 1  sinax  b dx   a cosax  b   c 1  sec ax  b dx  a tanax  b   c 2 1 e.g. i   sin 3 xdx   cos 3 x  c 3 ii   cos1  5 x dx   1 sin 1  5 x   c 5 iii   sec 2  x dx   2
Integrating Trig 1  cosax  b dx a  sin ax  b   c 1  sinax  b dx   a cosax  b   c 1  sec ax  b dx  a tanax  b   c 2 1 e.g. i   sin 3 xdx   cos 3 x  c 3 ii   cos1  5 x dx   1 sin 1  5 x   c 5  x dx  2 tan x   c iii   sec     2 2  2
 2 iv   sin 2 xdx  6
  2  1 cos 2 x  2 iv   sin 2 xdx      2  6 6
  2  1 cos 2 x  2 iv   sin 2 xdx      2  6 6 1     cos   cos  2 3
  2  1 cos 2 x  2 iv   sin 2 xdx      2  6 6 1     cos   cos  2 3 1 1    1   2 2 3  4
   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
   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 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 
   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
   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
   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 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
vii   sin 2 xdx
vii   sin 2 xdx cos2 
vii   sin xdx 2 cos2  cos 2  sin 2 
vii   sin xdx 2 cos2  cos 2  sin 2  1  1  2 sin 2   sin 2   1  cos 2  2
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
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
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   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
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
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
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
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  logcos x   c 1  log sec x  c
π 2 ix   cos x sin 7 xdx 0
π 2 ix   cos x sin 7 xdx u  sin x 0
π 2 ix   cos x sin 7 xdx u  sin x 0 du  cos xdx
π 2 ix   cos x sin 7 xdx u  sin x 0 du  cos xdx x  0 ,u  0  x ,u 1 2
π 2 ix   cos x sin 7 xdx u  sin x 0 1 du  cos xdx   u 7 du x  0 ,u  0 0  x ,u 1 2
π 2 ix   cos x sin 7 xdx u  sin x 0 1 du  cos xdx   u 7 du x  0 ,u  0 0  1 8  1 x ,u 1  u  2 8  0
π 2 ix   cos x sin 7 xdx u  sin x 0 1 du  cos xdx   u 7 du x  0 ,u  0 0  1 8  1 x ,u 1  u  2 8  0  18  0  1 8 1  8
π 2 ix   cos x sin 7 xdx u  sin x 0 1 du  cos xdx   u 7 du x  0 ,u  0 0  1 8  1 x  ,u 1  u  2 8  0  18  0  1 8 1  8 Exercise 14I; 2ace etc, 3ace etc, 4, 6, 8a, 9ac, 10a, 12ace, 13b(i), 14df, 15ace Exercise 14J; 2b, 3bfh, 4a, 5ac, 7, 9, 10, 13, 14, 21, 26

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12X1 T03 04 integrating trig functions

  • 2. Integrating Trig  cosax  b dx
  • 3. Integrating Trig 1  sin ax  b   c  cosax  b dx a
  • 4. Integrating Trig 1  sin ax  b   c  cosax  b dx a  sinax  b dx
  • 5. Integrating Trig 1  sin ax  b   c  cosax  b dx a 1  sinax  b dx   a cosax  b   c
  • 6. Integrating Trig 1  sin ax  b   c  cosax  b dx a 1  sinax  b dx   a cosax  b   c  sec 2 ax  b dx
  • 7. Integrating Trig 1  cosax  b dx a sin ax  b   c 1  sinax  b dx   a cosax  b   c 1  sec ax  b dx  a tanax  b   c 2
  • 8. Integrating Trig 1  cosax  b dx a sin ax  b   c 1  sinax  b dx   a cosax  b   c 1  sec ax  b dx  a tanax  b   c 2 e.g. i   sin 3 xdx
  • 9. Integrating Trig 1  cosax  b dx a  sin ax  b   c 1  sinax  b dx   a cosax  b   c 1  sec ax  b dx  a tanax  b   c 2 1 e.g. i   sin 3 xdx   cos 3 x  c 3
  • 10. Integrating Trig 1  cosax  b dx a  sin ax  b   c 1  sinax  b dx   a cosax  b   c 1  sec ax  b dx  a tanax  b   c 2 1 e.g. i   sin 3 xdx   cos 3 x  c 3 ii   cos1  5 x dx
  • 11. Integrating Trig 1  cosax  b dx a  sin ax  b   c 1  sinax  b dx   a cosax  b   c 1  sec ax  b dx  a tanax  b   c 2 1 e.g. i   sin 3 xdx   cos 3 x  c 3 ii   cos1  5 x dx   1 sin 1  5 x   c 5
  • 12. Integrating Trig 1  cosax  b dx a  sin ax  b   c 1  sinax  b dx   a cosax  b   c 1  sec ax  b dx  a tanax  b   c 2 1 e.g. i   sin 3 xdx   cos 3 x  c 3 ii   cos1  5 x dx   1 sin 1  5 x   c 5 iii   sec 2  x dx   2
  • 13. Integrating Trig 1  cosax  b dx a  sin ax  b   c 1  sinax  b dx   a cosax  b   c 1  sec ax  b dx  a tanax  b   c 2 1 e.g. i   sin 3 xdx   cos 3 x  c 3 ii   cos1  5 x dx   1 sin 1  5 x   c 5  x dx  2 tan x   c iii   sec     2 2  2
  • 14. 2 iv   sin 2 xdx  6
  • 15.  2  1 cos 2 x  2 iv   sin 2 xdx      2  6 6
  • 16.  2  1 cos 2 x  2 iv   sin 2 xdx      2  6 6 1     cos   cos  2 3
  • 17.  2  1 cos 2 x  2 iv   sin 2 xdx      2  6 6 1     cos   cos  2 3 1 1    1   2 2 3  4
  • 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
  • 25. vii   sin 2 xdx
  • 26. vii   sin 2 xdx cos2 
  • 27. vii   sin xdx 2 cos2  cos 2  sin 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  logcos x   c 1  log sec x  c
  • 37. π 2 ix   cos x sin 7 xdx 0
  • 38. π 2 ix   cos x sin 7 xdx u  sin x 0
  • 39. π 2 ix   cos x sin 7 xdx u  sin x 0 du  cos xdx
  • 40. π 2 ix   cos x sin 7 xdx u  sin x 0 du  cos xdx x  0 ,u  0  x ,u 1 2
  • 41. π 2 ix   cos x sin 7 xdx u  sin x 0 1 du  cos xdx   u 7 du x  0 ,u  0 0  x ,u 1 2
  • 42. π 2 ix   cos x sin 7 xdx u  sin x 0 1 du  cos xdx   u 7 du x  0 ,u  0 0  1 8  1 x ,u 1  u  2 8  0
  • 43. π 2 ix   cos x sin 7 xdx u  sin x 0 1 du  cos xdx   u 7 du x  0 ,u  0 0  1 8  1 x ,u 1  u  2 8  0  18  0  1 8 1  8
  • 44. π 2 ix   cos x sin 7 xdx u  sin x 0 1 du  cos xdx   u 7 du x  0 ,u  0 0  1 8  1 x  ,u 1  u  2 8  0  18  0  1 8 1  8 Exercise 14I; 2ace etc, 3ace etc, 4, 6, 8a, 9ac, 10a, 12ace, 13b(i), 14df, 15ace Exercise 14J; 2b, 3bfh, 4a, 5ac, 7, 9, 10, 13, 14, 21, 26