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11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
11 x1 t16 02 definite integral (2013)
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11 x1 t16 02 definite integral (2013)

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  • 1. Properties Of Definite Integral
  • 2. Properties Of Definite Integral 1 a b n 1 b x  x dx   n  1 a   n
  • 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. 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. 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. 5 a x n dx  0 b
  • 7. 5 a x n dx  0 b , if f  x   0 for a  x  b
  • 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. 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. 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. 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. 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. 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. 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. 2 e.g. (i)  6 x 2 dx 1
  • 16. 2 e.g. (i)  6 x 2 dx 1 2 1 x 3   6  3 1 
  • 17. 2 e.g. (i)  6 x 2 dx 1 2 1 x 3   6  3 1   223  13   14
  • 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. 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. 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. 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. 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. 2 iii   sin 5 xdx 2
  • 24. 2 iii   sin 5 xdx  0 2 odd function 5  odd function
  • 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. 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. 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. 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. 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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