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# 12 x1 t02 02 integrating exponentials (2014)

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### 12 x1 t02 02 integrating exponentials (2014)

1. 1. Integrating Exponentials
2. 2. Integrating Exponentials e ax dx  
3. 3. Integrating Exponentials 1 ax  e dx  a e  c ax
4. 4. Integrating Exponentials 1 ax  e dx  a e  c ax  f  x e f  x dx 
5. 5. Integrating Exponentials 1 ax  e dx  a e  c ax  f  x e f  x dx  e f  x   c
6. 6. Integrating Exponentials 1 ax  e dx  a e  c ax  e.g. i   e5 x dx f  x e f  x dx  e f  x   c
7. 7. Integrating Exponentials 1 ax  e dx  a e  c ax  e.g. i   e5 x dx 1  e5 x  c 5 f  x e f  x dx  e f  x   c
8. 8. Integrating Exponentials 1 ax  e dx  a e  c ax  e.g. i   e5 x dx 1  e5 x  c 5 f  x e f  x dx  e f  x   c OR e5 x dx  1   5e5 x dx 5
9. 9. Integrating Exponentials 1 ax  e dx  a e  c ax  e.g. i   e5 x dx 1  e5 x  c 5 f  x e f  x dx  e f  x   c OR e5 x dx  1   5e5 x dx 5 1  e5 x  c 5
10. 10. ii   xe dx x2
11. 11. 1 x2 ii   xe dx   2 xe dx 2 x2
12. 12. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2
13. 13. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2 iii   e9 x 5 dx  1  9e9 x 5 dx 9
14. 14. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2 iii   e9 x 5 dx  1  9e9 x 5 dx 9 1 9 x 5  e c 9
15. 15. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2 iv   e x dx iii   e9 x 5 dx  1  9e9 x 5 dx 9 1 9 x 5  e c 9
16. 16. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2 iv   x 2 e x dx   e dx iii   e9 x 5 dx  1  9e9 x 5 dx 9 1 9 x 5  e c 9
17. 17. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2 iv   x 2 e x dx   e dx x 1  2  e 2 dx 2 iii   e9 x 5 dx  1  9e9 x 5 dx 9 1 9 x 5  e c 9
18. 18. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2 iv   x 2 e x dx   e dx x 1  2  e 2 dx 2 x 2  2e  c  2 ex  c iii   e9 x 5 dx  1  9e9 x 5 dx 9 1 9 x 5  e c 9
19. 19. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2 iv   x 2 e x dx   e dx x 1  2  e 2 dx 2 x 2  2e  c  2 ex  c iii   e9 x 5 dx  1  9e9 x 5 dx 9 1 9 x 5  e c 9 v   e x  1e x  3dx
20. 20. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2 iv   x 2 e x dx   e dx x 1  2  e 2 dx 2 x 2  2e  c  2 ex  c iii   e9 x 5 dx  1  9e9 x 5 dx 9 1 9 x 5  e c 9 v   e  1e  3dx   e  2e  3dx x x 2x x
21. 21. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2 iv   x 2 e x dx   e dx x 1  2  e 2 dx 2 x 2  2e  c  2 ex  c iii   e9 x 5 dx  1  9e9 x 5 dx 9 1 9 x 5  e c 9 v   e  1e  3dx   e  2e  3dx x x 2x x 1 2x  e  2e x  3 x  c 2
22. 22. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2 iv   x 2 e x dx   e dx x 1  2  e 2 dx 2 x 2  2e  c  2 ex  c iii   e9 x 5 dx  1  9e9 x 5 dx 9 1 9 x 5  e c 9 v   e  1e  3dx   e  2e  3dx x x 2x x 1 2x  e  2e x  3 x  c 2 e5 x  e x vi   2 x dx e
23. 23. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2 iv   x 2 e x dx   e dx x 1  2  e 2 dx 2 x 2  2e  c  2 ex  c iii   e9 x 5 dx  1  9e9 x 5 dx 9 1 9 x 5  e c 9 v   e  1e  3dx   e  2e  3dx x x 2x x 1 2x  e  2e x  3 x  c 2 e5 x  e x vi   2 x dx e   e3 x  e  x dx
24. 24. 1 x2 ii   xe dx   2 xe dx 2 1 x2  e c 2 x2 iv   x 2 e x dx   e dx x 1  2  e 2 dx 2 x 2  2e  c  2 ex  c iii   e9 x 5 dx  1  9e9 x 5 dx 9 1 9 x 5  e c 9 v   e  1e  3dx   e  2e  3dx x x 2x x 1 2x  e  2e x  3 x  c 2 e5 x  e x vi   2 x dx e   e3 x  e  x dx 1 3x x  e e c 3
25. 25. vii  1 x e 2 x3 0 dx
26. 26. vii  1 x e 2 x3 0 1 dx 1 2 x3   3 x e dx 30
27. 27. vii  1 x e 2 x3 0 1 dx 1 2 x3   3 x e dx 30 1 x3 1  e 0 3  
28. 28. vii  1 x e 2 x3 0 1 dx 1 2 x3   3 x e dx 30 1 x3 1  e 0 3 1 1 0  e  e  3  
29. 29. vii  1 x e 2 x3 0 1 dx 1 2 x3   3 x e dx 30 1 x3 1  e 0 3 1 1 0  e  e  3 1  e  1 3  
30. 30. vii  1 x e 2 x3 0 1 dx 1 2 x3   3 x e dx 30 1 x3 1  e 0 3 1 1 0  e  e  3 1  e  1 3   viii   3x dx
31. 31. vii  1 x e 2 x3 0 1 dx 1 2 x3   3 x e dx 30 1 x3 1  e 0 3 1 1 0  e  e  3 1  e  1 3   viii   3x dx 3x  c log 3
32. 32. vii  1 x e 2 x3 0 1 dx 1 2 x3   3 x e dx 30 1 x3 1  e 0 3 1 1 0  e  e  3 1  e  1 3 viii   3x dx 3x  c log 3   Exercise 13C; 2 to 8 ace etc, 9, 10, 11, 13, 17 Exercise 13D; 2 to 18 evens, 21*