SME
Calculus II
Improper integrals
Use this for “Does the integral of 1 ÷ x³ from 1 to ∞ converge? If so, to what?”
Chart
Shaded: ∫ from 1 to 10 of f(x) dx = 0.9
This is an example. Type your own values to start a new problem.
∫
1
1
x
dx
=
Ib
∫
∞
1
1
x
dx
=
I
C
Converges or diverges
worked out (not typed)
Formulas
Ib
=
b
1 −
p
− 1
1 −
p
0.9
=
10
1 − 2
− 1
1 − 2
C
=
converges
if
p
>
1,
else
diverges
2 > 1, so converges
I
=
1
p
− 1
1
=
1
2 − 1
Assumptions
•
An improper integral is a limit of ordinary ones: ∫ from 1 to ∞ of f(x) dx is the limit of ∫ from 1 to b as b → ∞.
•
It converges when that limit is a finite number and diverges when the integrals to b grow without end.
•
An antiderivative of x⁻ᵖ is x¹⁻ᵖ ÷ (1 − p), or ln x when p = 1. The power p is more than 0.
Step-by-step
WE KNOW
p = 2, b = 10
FIND
integral from 1 to b (Ib), converges or diverges (C), limit as b → ∞ (I)
Step 1 · Find integral from 1 to b (Ib)
Use Ib = (b¹⁻ᵖ − 1) ÷ (1 − p)
Raise
the
power
−
p
by
one
to
1
−
p
and
divide
by
the
new
power.
Take
the
antiderivative’s
value
at
1
from
its
value
at
b
.
2 ≠ 1: the power rule
Ib
=
10
1 − 2
− 1
1 − 2
∫
x
-2
dx
=
x
-1
−1
+
C
10
∫
1
x
-2
dx
=
[
x
-1
−1
]
from
1
to
10
=
0.9
Ib = 0.9
Step 2 · Find converges or diverges (C)
Use C = converges if p > 1, else diverges
With
p
>
1
the
power
1
−
p
is
negative,
so
b
1 −
p
shrinks
to
0
as
b
grows:
the
integrals
to
b
level
off.
2 > 1: converges
C
=
converges
if
2
>
1,
else
diverges
C
=
converges
Step 3 · Find limit as b → ∞ (I)
Use I = 1 ÷ (p − 1)
As
b
→
∞,
b
1 −
p
→
0,
so
the
integral
to
b
approaches
0 − 1
1 −
p
=
1
p
− 1
.
I
=
1
2 − 1
I
=
1
Check
Put the numbers back in. Both sides should match.
0.9
=
10
1 − 2
− 1
1 − 2
✓
2 > 1, so converges ✓
1
=
1
2 − 1
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
More problem types
An integral that blows up at 0
Use this for “Does the integral of 1 ÷ ∜(x³) from 0 to 1 converge? If so, to what?” (p = 3/4).
An exponential decay to infinity
Use this for “Evaluate the integral of e⁻³ˣ from 0 to ∞, or show it diverges.”
The comparison test
Use this for “Show that the integral of 1 ÷ (x² + 9) from 1 to ∞ converges, and find its value” (c = 3).