11 x1 t16 02 definite integral (2013)

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11 x1 t16 02 definite integral (2013)

  1. 1. Properties Of Definite Integral
  2. 2. Properties Of Definite Integral 1 a b n 1 b x  x dx   n  1 a   n
  3. 3. Properties Of Definite Integral 1 a b n 1 b x  x dx   n  1 a   n 2 a kf  x dx k a f  x dx b b can only factorise constants 
  4. 4. Properties Of Definite Integral 1 a b n 1 b x  x dx   n  1 a   n 2 a kf  x dx k a f  x dx b b can only factorise constants  3 a  f  x   g  x dx  a f  x dx  a g  x dx b b b
  5. 5. Properties Of Definite Integral 1 a b n 1 b x  x dx   n  1 a   n can only factorise constants  2 a kf  x dx k a f  x dx b b 3 a  f  x   g  x dx  a f  x dx  a g  x dx b b b 4 a f  x dx  a f  x dx  c f  x dx b c b
  6. 6. 5 a x n dx  0 b
  7. 7. 5 a x n dx  0 b , if f  x   0 for a  x  b
  8. 8. 5 a x n dx  0 b 0 , if f  x   0 for a  x  b , if f  x   0 for a  x  b
  9. 9. 5 a x n dx  0 b , if f  x   0 for a  x  b , if f  x   0 for a  x  b 0 6 a f  x dx  a g  x dx b b
  10. 10. 5 a x n dx  0 b , if f  x   0 for a  x  b , if f  x   0 for a  x  b 0 6 a f  x dx  a g  x dx b b , if f  x   g  x  for a  x  b
  11. 11. 5 a x n dx  0 b , if f  x   0 for a  x  b , if f  x   0 for a  x  b 0 6 a f  x dx  a g  x dx b b 7 a f  x dx   b f  x dx b a , if f  x   g  x  for a  x  b
  12. 12. 5 a x n dx  0 b , if f  x   0 for a  x  b , if f  x   0 for a  x  b 0 6 a f  x dx  a g  x dx b b , if f  x   g  x  for a  x  b 7 a f  x dx   b f  x dx b a 8  f  x   0 , a a if f  x  is odd
  13. 13. 5 a x n dx  0 b , if f  x   0 for a  x  b , if f  x   0 for a  x  b 0 6 a f  x dx  a g  x dx b b , if f  x   g  x  for a  x  b 7 a f  x dx   b f  x dx b a a 8  f  x   0 , a a a a 0 if f  x  is odd 9  f  x   2 f  x  , if f  x  is even
  14. 14. 5 a x n dx  0 b , if f  x   0 for a  x  b , if f  x   0 for a  x  b 0 6 a f  x dx  a g  x dx b b , if f  x   g  x  for a  x  b 7 a f  x dx   b f  x dx b a a 8  f  x   0 , a a a a 0 if f  x  is odd 9  f  x   2 f  x  , if f  x  is even NOTE : odd  odd  even odd  even  odd even  even  even
  15. 15. 2 e.g. (i)  6 x 2 dx 1
  16. 16. 2 e.g. (i)  6 x 2 dx 1 2 1 x 3   6  3 1 
  17. 17. 2 e.g. (i)  6 x 2 dx 1 2 1 x 3   6  3 1   223  13   14
  18. 18. 2 e.g. (i)  6 x 2 dx 1 5 ii   3 xdx 0 2 1 x 3   6  3 1   223  13   14
  19. 19. 2 e.g. (i)  6 x 2 dx 1 2 1 x 3   6  3 1   223  13   14 5 5 0 0 1 3 ii   3 xdx   x dx
  20. 20. 2 e.g. (i)  6 x 2 dx 1 2 1 x 3   6  3 1   223  13   14 5 5 0 0 1 3 ii   3 xdx   x dx 5 3   x  4  0 4 3
  21. 21. 2 e.g. (i)  6 x 2 dx 1 2 1 x 3   6  3 1   223  13   14 5 5 0 0 1 3 ii   3 xdx   x dx 5 3   x  4  0 3 3 5  x x 0 4 4 3
  22. 22. 2 e.g. (i)  6 x 2 dx 1 2 1 x 3   6  3 1   223  13   14 5 5 0 0 1 3 ii   3 xdx   x dx 5 3   x  4  0 3 3 5  x x 0 4 3 3  5 5  0  4 153 5  4 4 3
  23. 23. 2 iii   sin 5 xdx 2
  24. 24. 2 iii   sin 5 xdx  0 2 odd function 5  odd function
  25. 25. 2 iii   sin 5 xdx  0 2 1 iv  x 3  2 x 2  x  1dx 1 odd function 5  odd function
  26. 26. 2 iii   sin 5 xdx  0 odd function 5  odd function 2 1 1 1 0 iv  x 3  2 x 2  x  1dx  2 2 x 2  1dx
  27. 27. 2 iii   sin 5 xdx  0 odd function 5  odd function 2 1 1 1 0 iv  x 3  2 x 2  x  1dx  2 2 x 2  1dx 1  2 x3  x  2 0 3 
  28. 28. 2 iii   sin 5 xdx  0 odd function 5  odd function 2 1 1 1 0 iv  x 3  2 x 2  x  1dx  2 2 x 2  1dx 1  2 x3  x  2 0 3   2 1 3  1  0  2    3  10  3
  29. 29. 2 iii   sin 5 xdx  0 odd function 5  odd function 2 1 1 1 0 iv  x 3  2 x 2  x  1dx  2 2 x 2  1dx 1  2 x3  x  2 0 3   2 1 3  1  0  2    3  10  3 Exercise 11C; 1bce, 2adf, 3ab (i, iii), 4bcf, 5, 6ac, 7df, 8b, 12b, 13*

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