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2.18. ∫
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2.19. ∫
dx
1 23 x + 3 x
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2.20. ∫
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2.25. ∫ (x - 2) ×(1+ 3 x - 2 )
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2.26.∫ cos2 x - 9sin 2 x
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2.27.∫ 2 cos2 x - 8sin 2 x
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30
Задание № 3
Вычислить несобственные интегралы (или установить их расходимость):
∞
3.01.∫ dx
x6
1
∞
dx
3.02. ∫ x2 + 2x + 3
2
∞
3.03. ∫ ln xxdx
2
0
dx
3.04. −∫∞ x2 + 4x + 5
0
3.05. ∫
dx
−∞ 4x2 + 4x + 2
0
3.06. ∫
dx
−∞9x2 + 6x + 2
∞ ln2 x dx
3.07. ∫
x
2
∞
dx
3.08. ∫2 (x −1)5
∞
dx
3.09. ∫3 (x − 2)4
∞
3.10. ∫
dx
2 x2 + 2x + 2
∞
xdx
3.11.∫ x2 +1
0
∞ x2 dx
3.12.∫ x3 +10
0
∞x3 dx
3.13.∫0 x4 + 5
∞x2 dx
3.14.∫ 3x3 + 7
0
∞ x3 dx
3.15.∫ 4x4 − 3
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31
∞ 1 + x2
3.19. ∫ dx x5
1
∞
arctg x
3.20. ∫ 1 + x2 dx
0
∞ ln3 x dx
3.21. ∫
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3.22. ∫ |
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3.25. ∫
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−∞4 + x2
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3.26. ∫
dx
−∞ x2 − 6x +10
∞
3.27. ∫e− x3 x2 dx
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3.28. ∫0 |
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32
Задание № 4
Вычислить несобственные интегралы (или установить их расходимость):
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4.01. ∫ |
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4.02. ∫ 2 − x2
1
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x dx
4.03. ∫ 3 + x
−3
2
4.04. ∫ dx
−1 5 x4
3
dx
4.05. ∫0 (x −1)3
1
2
4.06. ∫ dx x ln 2 x
0
3
dx
4.07. ∫0 (x −1)2
2
4.08. ∫ dx x ln 4 x
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4.09. ∫ |
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4.10. ∫ |
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4.11. ∫ctgx dx
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π
2
4.12. ∫ctg 2 x dx
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4.13. ∫ x ln x dx
0
5
dx
4.14. ∫0 (x − 2)6
e
4.15. ∫ |
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4.16. ∫2 3(5 − x)2
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4.17. ∫ |
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4.18. ∫sin |
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4.19. ∫ |
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4.20. ∫ |
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4.21. ∫cos |
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4.22. ∫ |
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4.23. ∫ |
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4.24.∫
0 9 - x2
22 + x
4.25.∫ dx
0 4 - x2
9
dx
4.26. ∫1 3(x -1)2
1
dx
4.27. ∫ x3 - 5x2
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4.29. ∫
dx
−2 3 - x2 - 2x
3
4.30. ∫
dx
2 6x - x2 - 8
34
Задание № 5
Вычислить площадь фигуры, ограниченной линиями:
y = x2 + x
y = x +1
y = −2x2 + 3x + 6
=x + 2y
y + 2x2 = 20 − 6x
y = 0
y = x2
lg x + lg y = 0= 05.04. y
=x 2
y = 2− x
5.05. x − 2 y + 2 = 0x − 2 = 0
y = 4x − x2
5.06.
y − x = 0
ln 5 − ln x − ln y = 0
5.07.
y = 6 − x
y = x3
5.08. y = 1
=
x 2x
y = x2 + 1
y = 3 − x2
y = 2 x
5.10.y = 4x + 2 = 0
ln x + ln y = ln 7
=
y 0
x − 3 = 0
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y = (x −1)2
y = x +1
35
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5.15. x 2 y 2 0
x = 0
y = 2x
5.16.y = 4xx −1 = 0
y = x2
y = 2 2x
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5.18. |
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5.20.y = 4 − 3xy = 0
y − x2 = 0
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36
y = log2 x
5.23. y = 0x = 8
y = tg x=
5.24. y 0
x = π3
y + x2 = 2x
y + x = 0y = e2 x
5.26.y = e x−2
x − 3 = 0
y − sin x = 0
5.27.= 2xy π
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5.30.y = 6 − 2xx = 0
37
Задание № 6
Вычислить площадь фигуры, ограниченной кривыми
x = 2(t − sin t)
6.01. y = 2(1− cost )
y = 0, 0 ≤ t ≤ 2π
6.02.ρ 2 = 2 cos 2θ
6.03.ρ = 4 cos 3ϕ
6.04.ρ = 3(1+ cosϕ )
6.05.ρ = 2(1− cosϕ )
6.06.ρ = 2(1− sinϕ )
6.07.ρ = α cos 2ϕ
6.08. Кардиоидой ρ = α (1 − cosϕ )
6.09. Между первым и вторым витками спирали ρ=2φ
6.10. ρ = 3ϕ, где π ≤ ϕ ≤ π
42
6.11.Цепной линией
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6.12. ρ = α cos 3ϕ
38
6.13. |
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6.14. |
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6.15.
0 ≤ ϕ ≤ 2π
6.16.ρ = 1 + sinϕ
6.17.ρ = (1+ cosϕ )
6.18.ρ = 2 cos 2ϕ
6.19.ρ = 2sin 2ϕρ = cosϕ
ρ = 2 cosϕ
ρ = cosϕ |
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6.23. ρ = 2 + 2 sinϕ
6.24. ρ = α (sinϕ + cosϕ )
39