Determine whether this improper integral converges or diverges. If it converges, evaluate it.
-1
⌠ dx
⎮ x3
⌡
-∞
(The only trouble spot is at –∞. Why isn't x = 0 a trouble spot?)
-1
⌠ dx
⎮ x3
⌡
-∞
⌠ dx
⎮ x3
⌡
-1
⌠ dx
⎮ x3
⌡
M
-1
lim ⌠ dx
= M→
3
−∞ ⎮
⌡ x
M
⌠x-3 dx
⌡
=
=
=
-1
lim ⌠ dx
3
M→ −∞ ⎮
⌡ x
M
=
Determine whether this improper integral converges or diverges. If it converges, evaluate it.
1
dx
⌠
⎮ (x - 1)2/3
⌡
0
(There is a trouble spot at x = 1.)
1
dx
⌠
⎮ (x - 1)2/3
⌡
0
dx
⌠
⎮ (x - 1)2/3
⌡
M
dx
⌠
⎮ (x - 1)2/3
⌡
0
M
dx
lim − ⌠
= M→1
⎮ (x - 1)2/3
⌡
0
-2/3
= ⌠
⌡(x - 1) dx
=
M
dx
lim − ⌠
⎮
M→1 ⌡ (x - 1)2/3
0
=
=
Determine whether this improper integral converges or diverges. If it converges, evaluate it.
π/2
2
⌠
⌡ tan x dx
0
(There is a trouble spot at π/2.)
M
lim
2
2
⌠
⌡ tan x dx = M→π/2− ⌠
⌡ tan x dx
0
0
π/2
⌠tan2x dx =
⌡
⌠sec2x dx
⌡
- ⌠dx
⌡
=
tanx - x + C
M
⌠ tan2x dx =
⌡
0
[tanM - M] - [tan0 - 0] =
M
lim − ⌠ tan2x dx =
M→π/2 ⌡
0
tanM - M
lim
M→π/2− [tanM - M] →
+∞ - π/2 → +∞
π/2
2
⌠
⌡ tan x dx diverges.
0
Determine whether this improper integral converges or diverges. If it converges, evaluate it.
π/2
⌠
⌡ tanx dx
0
(The trouble spot is at π/2)
π/2
⌠ tanx dx =
⌡
0
M
lim − ⌠ tanx dx
M→π/2 ⌡
0
dx =
⌠tanx
⌡
⌠ sinx dx
⎮cosx
⌡
=
ln⎥ cosx ⎢ + C
M
⌠ tanx dx
⌡
0
=
–ln ⎢cosM ⎢ + ln ⎢cos0 ⎢ =
–ln ⎢cosM ⎢ + ln1 =
–ln ⎢cosM ⎢
M
lim − ⌠ tanx dx =
M→π/2 ⌡
0
lim
M→π/2− –ln⎥cosM⎥
π/2
⌠
⌡ tanx dx diverges.
0
→
+∞
Determine whether this improper integral converges or diverges. If it converges, evaluate it.
∞
M
1
M
1
1
1
lim − ⌠
lim
lim
⌠
⌠ 1
⌠
= M→0
⎮ 3 x dx
⎮ 3 x dx + M→0+ ⎮ 3 x dx + M→∞ ⎮ 3 x dx
⌡
⌡
⌡
⌡
-1
M
1
-1
(There are trouble spots at 0 and ∞)
⌠ 1 dx = ⌠x-1/3 dx
⌡
⎮3 x
⌡
3
= 2 x2/3 + C
M
⌠ 1 dx = 3 (M2/3 - (-1)2/3 ) = 3 M2/3 - 3
⎮3 x
2
2
2
⌡
-1
1
⌠ 1 dx = 3 (12/3 - (M)2/3) = 3 - 3 M2/3
⎮3 x
2
2 2
⌡
M
M
⌠ 1 dx = 3 M2/3 - 3
⎮3 x
2
2
⌡
1
M
lim
⌠ 1
M→0− ⎮ 3 x dx
⌡
-1
lim − ⎛⎜ 3 M2/3 - 3 ⎞⎟ = – 3
= M→0
2⎠
2
⎝2
1
lim
lim + ⌠ 1 dx =
M→0+
M→0 ⎮ 3 x
⌡
M
M
lim
lim ⎛
⌠ 1
M→∞ ⎮ 3 x dx = M→∞ ⎜⎝
⌡
1
∞
⌠ 1 dx diverges
⎮3 x
⌡
-1
⎛ 3 - 3 M2/3 ⎞ = 3
⎜2 2
⎟
2
⎝
⎠
3 2/3 3 ⎞
-2⎟
2M
⎠
→
∞