Improper integrals

Print the chapter

Every lesson of the chapter, ready for paper. Exercise sets are lettered A, B, C… and answered at the end.

To save a PDF, choose “Save as PDF” in the print dialog.

1. Improper integrals over infinite intervals

Warm-up

Three quick questions from earlier lessons.

Discover

∫1bdxx2=1−1b\displaystyle\int_1^b \dfrac{dx}{x^2} = 1 - \dfrac{1}{b}. For b=10b = 10, 100100, 10001000 this is 0.90.9, 0.990.99, 0.9990.999. What happens as b→∞b \to \infty?

  1. It approaches 1
  2. It grows without bound

Key idea

An infinite interval is handled with a limit.

∫a∞f(x) dx=lim⁡b→∞∫abf(x) dx\displaystyle\int_a^{\infty} f(x)\,dx = \lim_{b \to \infty} \int_a^b f(x)\,dx. If the limit is a number, the integral converges to it; otherwise it diverges.

Worked example

∫0∞e−2x dx\displaystyle\int_0^{\infty} e^{-2x}\,dx.

∫0be−2x dx=1−e−2b2\displaystyle\int_0^b e^{-2x}\,dx = \dfrac{1 - e^{-2b}}{2}
Integrate to b.
lim⁡b→∞1−e−2b2=12\displaystyle\lim_{b \to \infty} \dfrac{1 - e^{-2b}}{2} = \dfrac{1}{2}
e^{−2b} → 0.
Converges to 12\tfrac{1}{2}
A finite limit.

Your turn, guided

Your turn: ∫1∞dxx3\displaystyle\int_1^{\infty} \dfrac{dx}{x^3}.

=12= \tfrac{1}{2}
[−1/(2x²)] from 1 to b → 0 + ½.

Practice ★

  1. Evaluate ∫1∞dxx4\displaystyle\int_1^{\infty} \dfrac{dx}{x^{ 4 } }.
  2. Evaluate ∫1∞dxx3\displaystyle\int_1^{\infty} \dfrac{dx}{x^{ 3 } }.
  1. Evaluate ∫0∞e−4x dx\displaystyle\int_0^{\infty} e^{-4x}\,dx.
  2. Evaluate ∫0∞e−3x dx\displaystyle\int_0^{\infty} e^{-3x}\,dx.

Watch out!

Practice ★★

  1. What is ∫9∞dxx\displaystyle\int_{ 9 }^{\infty} \dfrac{dx}{x}?
    1. It diverges: ln⁡b→∞\ln b \to \infty
    2. It converges, because 1x→0\dfrac{1}{x} \to 0
    3. It equals ln⁡9\ln 9
  1. Boss problem. Evaluate ∫0∞xe−1x dx\displaystyle\int_0^{\infty} x e^{-1x}\,dx.

Exit ticket

Three questions to check you have got it.

  1. Evaluate ∫1∞dxx4\displaystyle\int_1^{\infty} \dfrac{dx}{x^{ 4 } }.
  2. Evaluate ∫1∞dxx3\displaystyle\int_1^{\infty} \dfrac{dx}{x^{ 3 } }.
  3. Evaluate ∫1∞dxx2\displaystyle\int_1^{\infty} \dfrac{dx}{x^{ 2 } }.

Summary card

Infinite intervals

  1. Replace ∞\infty by bb.
  2. Integrate.
  3. Take lim⁡b→∞\lim_{b \to \infty}: a number means convergence.

Watch out: f→0f \to 0 does not guarantee convergence.

2. Improper integrals with infinite discontinuities

Warm-up

Three quick questions from earlier lessons.

Discover

1x\dfrac{1}{\sqrt{x}} blows up at x=0x = 0. Yet ∫t1dxx=2−2t\displaystyle\int_t^1 \dfrac{dx}{\sqrt{x}} = 2 - 2\sqrt{t}. What happens as t→0+t \to 0^+?

  1. It approaches 2
  2. It diverges because 1x\tfrac{1}{\sqrt{x}} is unbounded

Key idea

Approach a vertical asymptote with a limit.

If ff is unbounded at aa: ∫abf(x) dx=lim⁡t→a+∫tbf(x) dx\displaystyle\int_a^b f(x)\,dx = \lim_{t \to a^+} \int_t^b f(x)\,dx. If the asymptote is inside [a,b][a, b], split there; the integral converges only if both parts do.

Worked example

∫08dxx3\displaystyle\int_0^8 \dfrac{dx}{\sqrt[3]{x}}.

∫t8x−1/3 dx=6−32t2/3\displaystyle\int_t^8 x^{-1/3}\,dx = 6 - \tfrac{3}{2}t^{2/3}
∫x^(−1/3) dx = (3/2)x^(2/3).
→6\to 6
t → 0⁺.

Your turn, guided

Your turn: ∫04dxx\displaystyle\int_0^4 \dfrac{dx}{\sqrt{x}}.

=4= 4
2√x from t to 4, t → 0.

Practice ★

  1. Evaluate ∫025dxx\displaystyle\int_0^{ 25 } \dfrac{dx}{\sqrt{x} }.
  2. Evaluate ∫01dxx\displaystyle\int_0^{ 1 } \dfrac{dx}{\sqrt{x} }.
  3. Evaluate ∫016dxx\displaystyle\int_0^{ 16 } \dfrac{dx}{\sqrt{x} }.

Watch out!

Practice ★★

  1. What is ∫−22dxx2\displaystyle\int_{-2}^{ 2 } \dfrac{dx}{x^2}?
    1. It diverges: 1x2\dfrac{1}{x^2} is unbounded at x=0x = 0
    2. [−1x]−22=−22\left[-\dfrac{1}{x}\right]_{-2}^{ 2 } = -\dfrac{2}{ 2 }
    3. 0, by symmetry
  1. Evaluate ∫018ln⁡x dx\displaystyle\int_0^1 8\ln x\,dx.
  2. Evaluate ∫015ln⁡x dx\displaystyle\int_0^1 5\ln x\,dx.

Exit ticket

Three questions to check you have got it.

  1. Evaluate ∫025dxx\displaystyle\int_0^{ 25 } \dfrac{dx}{\sqrt{x} }.
  2. Evaluate ∫01dxx\displaystyle\int_0^{ 1 } \dfrac{dx}{\sqrt{x} }.
  3. Evaluate ∫016dxx\displaystyle\int_0^{ 16 } \dfrac{dx}{\sqrt{x} }.

Summary card

Infinite discontinuities

  • Look for asymptotes inside [a,b][a, b] before integrating.
  • Replace the bad end by tt; take the one-sided limit.

Watch out: never apply the FTC across an asymptote.

3. Convergence of improper integrals: the p-test

Warm-up

Three quick questions from earlier lessons.

Discover

∫1∞dxx\displaystyle\int_1^{\infty} \dfrac{dx}{x} diverges and ∫1∞dxx2\displaystyle\int_1^{\infty} \dfrac{dx}{x^2} converges. What about ∫1∞dxx3\displaystyle\int_1^{\infty} \dfrac{dx}{x^3}?

  1. It converges
  2. It diverges

Key idea

Powers of xx decide convergence at once.

p-test: ∫1∞dxxp\displaystyle\int_1^{\infty} \dfrac{dx}{x^p} converges exactly when p>1p > 1 (to 1p−1\tfrac{1}{p - 1}). ∫01dxxp\displaystyle\int_0^1 \dfrac{dx}{x^p} converges exactly when p<1p < 1 (to 11−p\tfrac{1}{1 - p}). For p=1p = 1 both diverge.

Worked example

Classify three integrals.

∫1∞x−3/2 dx\displaystyle\int_1^{\infty} x^{-3/2}\,dx converges, to 22
p = 3/2 > 1 at ∞.
∫01x−1/2 dx\displaystyle\int_0^1 x^{-1/2}\,dx converges, to 22
p = 1/2 < 1 near 0.
∫1∞x−1/2 dx\displaystyle\int_1^{\infty} x^{-1/2}\,dx diverges
p = 1/2 ≤ 1 at ∞.

Your turn, guided

Your turn: the value of ∫1∞dxx4\displaystyle\int_1^{\infty} \dfrac{dx}{x^4}.

13\tfrac{1}{3}
1/(p − 1) with p = 4.

Practice ★

  1. For which pp does ∫2∞dxxp\displaystyle\int_{ 2 }^{\infty} \dfrac{dx}{x^p} converge?
    1. p>1p > 1
    2. p≥1p \ge 1
    3. p<1p < 1
  1. Evaluate ∫3∞5x3 dx\displaystyle\int_{ 3 }^{\infty} \dfrac{ 5 }{x^3}\,dx.
  2. Evaluate ∫4∞1x3 dx\displaystyle\int_{ 4 }^{\infty} \dfrac{ 1 }{x^3}\,dx.

Watch out!

Practice ★★

  1. Which improper integrals converge? Tick all that do.
    1. ∫1∞3x2 dx\displaystyle\int_1^{\infty} \dfrac{ 3 }{x^2}\,dx
    2. ∫013x dx\displaystyle\int_0^1 \dfrac{ 3 }{\sqrt{x} }\,dx
    3. ∫0∞e−3x dx\displaystyle\int_0^{\infty} e^{-3x}\,dx
    4. ∫1∞3x dx\displaystyle\int_1^{\infty} \dfrac{ 3 }{\sqrt{x} }\,dx
    5. ∫013x2 dx\displaystyle\int_0^1 \dfrac{ 3 }{x^2}\,dx
  2. Which improper integrals converge? Tick all that do.
    1. ∫1∞5x2 dx\displaystyle\int_1^{\infty} \dfrac{ 5 }{x^2}\,dx
    2. ∫015x dx\displaystyle\int_0^1 \dfrac{ 5 }{\sqrt{x} }\,dx
    3. ∫0∞e−5x dx\displaystyle\int_0^{\infty} e^{-5x}\,dx
    4. ∫1∞5x dx\displaystyle\int_1^{\infty} \dfrac{ 5 }{\sqrt{x} }\,dx
    5. ∫015x2 dx\displaystyle\int_0^1 \dfrac{ 5 }{x^2}\,dx

Exit ticket

Three questions to check you have got it.

  1. Evaluate ∫3∞5x3 dx\displaystyle\int_{ 3 }^{\infty} \dfrac{ 5 }{x^3}\,dx.
  2. Evaluate ∫4∞1x3 dx\displaystyle\int_{ 4 }^{\infty} \dfrac{ 1 }{x^3}\,dx.
  3. Evaluate ∫4∞4x3 dx\displaystyle\int_{ 4 }^{\infty} \dfrac{ 4 }{x^3}\,dx.

Summary card

The p-test

  • ∫1∞x−p dx\int_1^{\infty} x^{-p}\,dx: converges for p>1p > 1.
  • ∫01x−p dx\int_0^1 x^{-p}\,dx: converges for p<1p < 1.
  • p=1p = 1: both diverge.

Watch out: the condition flips between ∞\infty and 00.

Answers: Improper integrals

1. Improper integrals over infinite intervals

A

  1. 13\frac{1}{3}
  2. 12\frac{1}{2}

B

  1. 14\frac{1}{4}
  2. 13\frac{1}{3}

C

  1. It diverges: ln⁡b→∞\ln b \to \infty

D

  1. 11

E

  1. 13\frac{1}{3}
  2. 12\frac{1}{2}
  3. 11

2. Improper integrals with infinite discontinuities

A

  1. 1010
  2. 22
  3. 88

B

  1. It diverges: 1x2\dfrac{1}{x^2} is unbounded at x=0x = 0

C

  1. −8-8
  2. −5-5

D

  1. 1010
  2. 22
  3. 88

3. Convergence of improper integrals: the p-test

A

  1. p>1p > 1

B

  1. 518\frac{5}{18}
  2. 132\frac{1}{32}

C

  1. ∫1∞3x2 dx\displaystyle\int_1^{\infty} \dfrac{ 3 }{x^2}\,dx · ∫013x dx\displaystyle\int_0^1 \dfrac{ 3 }{\sqrt{x} }\,dx · ∫0∞e−3x dx\displaystyle\int_0^{\infty} e^{-3x}\,dx
  2. ∫1∞5x2 dx\displaystyle\int_1^{\infty} \dfrac{ 5 }{x^2}\,dx · ∫015x dx\displaystyle\int_0^1 \dfrac{ 5 }{\sqrt{x} }\,dx · ∫0∞e−5x dx\displaystyle\int_0^{\infty} e^{-5x}\,dx

D

  1. 518\frac{5}{18}
  2. 132\frac{1}{32}
  3. 18\frac{1}{8}