Integrali generalizzati - unibo.itcinti/pdf/analisiT1/integrali.generalizz.pdf · Integrali...
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Integrali generalizzati Stabilire se i seguenti integrali generalizzati sono convergenti: 1. Z +∞ 0 ln(1 + x) x 2 dx; [non converge] 2. Z 1 0 x x - 1 x dx; [converge] 3. Z 1 0 ln(1 + x 2 ) sin x - sinh x dx; [non converge] 4. Z π 4 0 cosh 2 x - 1 arcsinh x - x dx; [converge] 5. Z e 2 ln ln x (e - x) 3 2 dx; [converge] 6. Z +∞ 0 arctan x - arctan x 2 x +1 dx; [converge] 7. Z +∞ 1 1+ 1 x x - e dx; [non converge] 8. Z 1 0 sin x x α √ 1 - x dx, α> 0; [converge ⇔ 0 <α< 2] 9. Z 1 0 e x tan(x 3 ) 3 √ 1+ x α - 1 dx, α> 0; [converge ⇔ 0 <α< 4] 10. Z +∞ 0 arctan x x α dx, α> 0; [converge ⇔ 1 <α< 2] 11. Z +∞ 0 x α (x + sin x) x - sin x dx, α> 0; [@ α per cui converga] 12. Z +∞ 0 x αx | sinh x - cosh 2 x| x 4α dx, α ∈ R ; [converge ⇔ α< 0] 13. Z 1 0 x ln 1+ x α 1 - x dx, α> 0; [converge ∀ α> 0] 14. Z +∞ 0 sin 2 x +1 - cos x x α dx, α> 0; [converge ⇔ 1 <α< 3] 1
Transcript of Integrali generalizzati - unibo.itcinti/pdf/analisiT1/integrali.generalizz.pdf · Integrali...
Integrali generalizzati
Stabilire se i seguenti integrali generalizzati sono convergenti:
1.
∫ +∞
0
ln(1 + x)
x2dx; [non converge]
2.
∫ 1
0
xx− 1
xdx; [converge]
3.
∫ 1
0
ln(1 + x2)
sinx− sinhxdx; [non converge]
4.
∫ π4
0
cosh2x− 1
arcsinhx− xdx; [converge]
5.
∫ e
2
ln lnx
(e− x) 32
dx; [converge]
6.
∫ +∞
0
(arctanx− arctan
x2
x+ 1
)dx; [converge]
7.
∫ +∞
1
((1 +
1
x
)x− e)
dx; [non converge]
8.
∫ 1
0
sinx
xα√
1− xdx, α > 0 ; [converge⇔ 0 < α < 2]
9.
∫ 1
0
ex tan(x3)3√
1 + xα − 1dx, α > 0 ; [converge⇔ 0 < α < 4]
10.
∫ +∞
0
arctanx
xαdx, α > 0 ; [converge⇔ 1 < α < 2]
11.
∫ +∞
0
xα(x+ sinx)
x− sinxdx, α > 0 ; [@α per cui converga]
12.
∫ +∞
0
xαx| sinhx− cosh2 x|x4α
dx, α ∈ R ; [converge⇔ α < 0]
13.
∫ 1
0
x ln1 + xα
1− xdx, α > 0 ; [converge ∀α > 0]
14.
∫ +∞
0
sin2x+ 1− cosx
xαdx, α > 0 ; [converge⇔ 1 < α < 3]
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15.
∫ +∞
0
(√1 + x−
√x)α
dx, α > 0 ; [converge⇔ α > 2]
16.
∫ +∞
π2
√| cosx|(
x− π2
)α(x
32 + 3
2
) dx, α ∈ R ;[converge⇔ −1
2< α < 3
2
]17.
∫ 1
0
sin(√x)
(1− x2
x
)αdx, α ∈ R ;
[converge⇔ −1 < α < 3
2
]18.
∫ +∞
2
e1xα
(x3 − 8)αdx, α > 0 ;
[converge⇔ 1
3< α < 1
]19.
∫ +∞
0
|x− sinx|(x+ x3)α
dx, α > 0 ;[converge⇔ 2
3< α < 4
]20.
∫ +∞
0
√x ln(1 + xα)
1 + x2dx, α > 0 ; [converge ∀α > 0]
21.
∫ 1
0
1(√x(1 + x)−
√x)α dx, α > 0 ;
[converge⇔ 0 < α < 2
3
]
22.
∫ +∞
1
(x+ 1)32
(e
1x2 − 1
)αdx, α > 0 ;
[converge⇔ α > 5
4
]23.
∫ +∞
0
ln4(1 + x) eαx3
(ex3− 1)α(1 + x)αdx, α > 0 ;
[converge⇔ 1 < α < 5
3
]24.
∫ +∞
0
(ex − 1)α
e3x arctan( 3√x)(x2 + x3)
dx, α > 0 ;[converge⇔ 4
3< α ≤ 3
]25.
∫ +∞
0
ln(1 + xα)
sinh(x6)(x2 + 6√x)
dx, α > 0 .[converge⇔ α > 31
6
]
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