Интегрирование по частям в неопределенном интеграле
Методом интегрирования по частям найдите интегралы:
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∫(5x −3)sin 2xdx . |
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∫(x +1)ln xdx . |
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∫xe−3 xdx . |
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∫x3x dx . |
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∫arctg2x dx . |
6. |
∫x2 cos3x dx . |
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∫x arcctgx dx . |
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∫arcsin |
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8. |
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dx . |
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∫ln2 xdx . |
10. |
∫ |
ln x |
dx . |
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11. ∫(x2 −5x −3)sin (x +1)dx . |
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12. |
∫ln (1 + x2 )dx . |
Интегрирование рациональных функций
Укажите, какие из указанных рациональных дробей являются правильными. Выделите целую часть в неправильных дробях:
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2x |
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7x2 +1. |
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x + |
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x +5 |
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x2 −1 |
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3x2 |
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5. |
x2 |
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6. |
x5 |
− 4x3 |
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x3 |
+1 |
x +3 |
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− x +1 |
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Найдите неопределенные интегралы от простейших рациональных функций (дробей):
7. ∫ |
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8. ∫ |
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(2x −1)2 |
3x +5 |
10. ∫ |
3 − 4x |
11. ∫ |
x − 2 |
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dx . |
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2x2 −3x − 2 |
x2 −7x +12 |
9. ∫ |
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5 |
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2x2 + 6x + 5 |
12. ∫ |
8 −3x |
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2x2 + x −1 |
Разложите дробь на простейшие, не находя коэффициентов разложения:
13.3x2 + 2x −5 .
x(x −1)3
x+1
15.(x −1)2 (x2 + x +1).
17. (x2 −5x + 4)(7 x2 −5x + 7).
x + 2
14. (x2 −1)(x2 + 2).
x3 + x
16. x4 −81 .
x − 2
18. x2 (x2 + 4)2 .
19. |
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2x + 3 |
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20. |
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x3 + x2 |
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(x2 + 4x + 3)(x2 − x + 3) |
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(x4 −16)(x3 +8) |
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21. |
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x2 +3x +1 |
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4x3 + 4x2 + 7x)2 |
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Найдите интегралы от рациональных функций: |
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∫ |
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x2 |
+3x +5 |
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∫ |
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xdx |
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24. ∫ |
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x5 + x4 −8 |
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22. |
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dx . |
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23. |
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dx . |
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x + 2 |
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(x + 2)(2x +1) |
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x3 − 4x |
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x2 + |
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26. |
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27. |
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x3 + 2 |
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∫x |
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−5x |
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+ 6x |
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∫ |
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∫x |
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− x |
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x −1 x |
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28. |
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∫ |
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29. |
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∫ |
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3x +1 |
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x(x2 + 4) |
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(x2 + 6x + 9)(x −5) |
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30. |
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∫ |
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7x −15 |
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∫ |
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x5 |
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32. ∫ |
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x2dx |
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x3 − 2x2 + 5x |
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x4 −1 |
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1 − x4 |
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x3 |
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33. |
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dx |
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34. |
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dx . |
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∫x |
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∫ |
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−3x |
+ 2 |
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−1 |
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1 + x |
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− 2x |
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∫ |
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1 − x |
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1 − x |
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= ( |
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1 + x |
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35. |
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dx, |
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= |
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− |
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7 |
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+ x |
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+ x |
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+ x |
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+ x |
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x |
1 + x |
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x 1 |
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x 1 |
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Метод рационализации: интегрирование функций, рационально зависящих от тригонометрических
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Найдите интегралы с помощью универсальной тригонометри- |
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ческой подстановки: |
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1. |
∫ |
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dx |
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2. |
∫ |
dx |
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3. |
∫ |
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dx |
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+ cos x |
4 −3sin x |
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+sin x + cos x |
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4. |
∫ |
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2 + cos x dx . |
5. |
∫ |
dx |
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∫ |
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2sin x −cos x +5 |
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3 −sin x |
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sin x |
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Найдите интегралы с помощью подходящей подстановки: |
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∫ |
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sin x |
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∫ |
cos3 x |
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∫sin |
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7. |
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dx . |
8. |
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dx . |
9. |
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xcos |
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x dx . |
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3 −sin2 x |
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+ cos x |
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10. |
∫ |
2sin2 x |
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12. ∫sin |
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2xcos 2xdx. |
cos |
2sin |
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∫ |
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∫ |
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sin x |
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cos2 x |
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dx. |
15. ∫sin6 x dx . |
4 −3sin2 x |
sin2 x + 6cos2 x |
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Найдите интегралы: |
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x |
cos 7x dx . |
16. |
∫cos3xsin 5x dx . |
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17. |
∫cos |
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18. |
∫sin 3x sin8x dx . |
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19. |
∫sin5 xdx . |
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20. |
∫cos4 2x dx . |
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21. |
∫sin2 xcos4 x dx . |
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22. ∫tg4 |
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dx . |
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23. |
∫ |
tg x |
dx . |
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1 +sin2 x |
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cos3 x |
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dx |
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24. |
∫sin6 x dx . |
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25. |
∫ |
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cos2 xsin4 x |
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dx |
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1 −sin x + cos x |
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26. |
∫ |
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27. |
∫1 +sin x −cos x dx . |
sin2 x +3sin xcos x −cos2 x |
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∫ |
1− tg x |
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∫ |
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28. |
1+ tg x dx . |
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29. |
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sin4 x + cos4 x |
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Метод рационализации: интегрирование простейших иррациональностей
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Найдите интегралы: |
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1. ∫ |
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dx |
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2. ∫ |
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dx |
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+ 6x + |
5 |
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− 4x + |
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4. ∫ |
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3x +1 |
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dx . |
5. ∫ |
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5x +3 |
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dx . |
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x |
2 |
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+ 2x + |
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− x +1 |
7. ∫ x (2 + x )6dx . |
8. ∫ |
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3 |
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− 2 x |
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10. ∫ |
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11. ∫ |
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xdx |
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2 4 |
x + |
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x −3)6 |
13. ∫ |
2 + 6 |
x |
dx . |
14. ∫ |
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x +1 |
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dx . |
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x + |
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3x −1 + |
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3. |
∫ |
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2x − x |
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6. |
∫ |
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x +3 |
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x +3 + |
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9. ∫ |
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1 −3x − |
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12. |
∫ |
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xdx |
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3 |
(5x − 2) |
2 |
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15. |
∫ |
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( |
2 + x )3 |
