Stewart 8th Section 7.8: Improper Integrals

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Stewart 8th Section 7.8: Improper Integrals 0/92
1 Exercise - Identifying Improper Integrals · Level 1
Explain why each of the following integrals is improper.
(a) \(\displaystyle\int_{1}^{2} \dfrac{x}{x-1} d x\)
(b) \(\displaystyle\int_{0}^{\infty} \dfrac{1}{1+x^3} d x\)
(c) \(\displaystyle\int_{-\infty}^\infty x^2 e^{-x^2} d x\)
(d) \(\displaystyle\int_{0}^{\dfrac{\pi}{4}} \cot x d x\)

Enter your answer directly below each part above.

2 Exercise - Identifying Improper Integrals · Level 1
Which of the following integrals are improper? Why?
(a) \(\displaystyle\int_{0}^{\dfrac{\pi}{4}} \tan x d x\)
(b) \(\displaystyle\int_{0}^{\pi} \tan x d x\)
(c) \(\displaystyle\int_{-1}^1 \dfrac{d x}{x^2 - x - 2}\)
(d) \(\displaystyle\int_{0}^{\infty} e^{-x^3} d x\)

Enter your answer directly below each part above.

3 Exercise - Area Under Curve · Level 2
Find the area under the curve \(y = 1/x^3\) from \(x = 1\) to \(x = t\) and evaluate it for \(t = 10, 100\), and \(1000\). Then find the total area under this curve for \(x \geq 1\).
4 Exercise - Comparing Convergence Behavior · Level 2
(a) Graph the functions \(f(x) = 1/x^{1.1}\) and \(g(x) = 1/x^{0.9}\) in the viewing rectangles \([0, 10]\) by \([0, 1]\) and \([0, 100]\) by \([0, 1]\).
(b) Find the areas under the graphs of \(f\) and \(g\) from \(x = 1\) to \(x = t\) and evaluate for \(t = 10, 100, 10^4, 10^6, 10^{10}, 10^{20}\).
(c) Find the total area under each curve for \(x \geq 1\), if it exists.

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5 Exercise - Convergence Test · Level 2
\( \displaystyle\int_{3}^{\infty} \dfrac{1}{(x-2)^{\dfrac{3}{2}}} d x \)
6 Exercise - Convergence Test · Level 2
\( \displaystyle\int_{0}^{\infty} \dfrac{1}{\sqrt[4]{1+x}} d x \)
7 Exercise - Convergence Test · Level 2
\( \displaystyle\int_{-\infty}^0 \dfrac{1}{3-4x} d x \)
8 Exercise - Convergence Test · Level 2
\( \displaystyle\int_{1}^{\infty} \dfrac{1}{(2x+1)^3} d x \)
9 Exercise - Convergence Test · Level 2
\( \displaystyle\int_{2}^{\infty} e^{-5p} d p \)
10 Exercise - Convergence Test · Level 2
\( \displaystyle\int_{-\infty}^0 2^r d r \)
11 Exercise - Convergence Test · Level 2
\( \displaystyle\int_{0}^{\infty} \dfrac{x^2}{\sqrt{1+x^3}} d x \)
12 Exercise - Convergence Test · Level 2
\( \displaystyle\int_{-\infty}^\infty (y^3 - 3y^2) d y \)
13 Exercise - Convergence Test · Level 2
\( \displaystyle\int_{-\infty}^\infty x e^{-x^2} d x \)
14 Exercise - Convergence Test · Level 2
\( \displaystyle\int_{1}^{\infty} \dfrac{e^{-\dfrac{1}{x}}}{x^2} d x \)
15 Exercise - Convergence Test · Level 2
\( \displaystyle\int_{0}^{\infty} \sin^2 \alpha d \alpha \)
16 Exercise - Convergence Test · Level 2
\( \displaystyle\int_{0}^{\infty} \sin \theta e^{\cos \theta} d \theta \)
17 Exercise - Convergence Test · Level 2
\( \displaystyle\int_{1}^{\infty} \dfrac{1}{x^2 + x} d x \)
18 Exercise - Convergence Test · Level 2
\( \displaystyle\int_{2}^{\infty} \dfrac{d v}{v^2 + 2v - 3} \)
19 Exercise - Convergence Test · Level 2
\( \displaystyle\int_{-\infty}^0 z e^{2z} d z \)
20 Exercise - Convergence Test · Level 2
\( \displaystyle\int_{2}^{\infty} y e^{-3y} d y \)
21 Exercise - Convergence Test · Level 2
\( \displaystyle\int_{1}^{\infty} \dfrac{\ln x}{x} d x \)
22 Exercise - Convergence Test · Level 2
\( \displaystyle\int_{1}^{\infty} \dfrac{\ln x}{x^2} d x \)
23 Exercise - Convergence Test · Level 2
\( \displaystyle\int_{-\infty}^0 \dfrac{z}{z^4 + 4} d z \)
24 Exercise - Convergence Test · Level 2
\( \displaystyle\int_{e}^{\infty} \dfrac{1}{x (\ln x)^2} d x \)
25 Exercise - Convergence Test · Level 2
\( \displaystyle\int_{0}^{\infty} e^{-\sqrt{y}} d y \)
26 Exercise - Convergence Test · Level 2
\( \displaystyle\int_{1}^{\infty} \dfrac{d x}{\sqrt{x} + x \sqrt{x}} \)
27 Exercise - Type 2 Convergence · Level 1
\( \displaystyle\int_{0}^{1} \dfrac{1}{x} d x \)
28 Exercise - Type 2 Convergence · Level 2
\( \displaystyle\int_{0}^{5} \dfrac{1}{\sqrt[3]{5-x}} d x \)
29 Exercise - Type 2 Convergence · Level 2
\( \displaystyle\int_{-2}^{14} \dfrac{d x}{\sqrt[4]{x+2}} \)
30 Exercise - Type 2 Convergence · Level 2
\( \displaystyle\int_{-1}^2 \dfrac{x}{(x+1)^2} d x \)
31 Exercise - Type 2 Convergence · Level 2
\( \displaystyle\int_{-2}^3 \dfrac{1}{x^4} d x \)
32 Exercise - Type 2 Convergence · Level 1
\( \displaystyle\int_{0}^{1} \dfrac{d x}{\sqrt{1 - x^2}} \)
33 Exercise - Type 2 Convergence · Level 2
\( \displaystyle\int_{0}^{9} \dfrac{1}{\sqrt[3]{x-1}} d x \)
34 Exercise - Type 2 Convergence · Level 2
\( \displaystyle\int_{0}^{5} \dfrac{w}{w-2} d w \)
35 Exercise - Type 2 Convergence · Level 2
\( \displaystyle\int_{0}^{\dfrac{\pi}{2}} \tan^2 \theta d \theta \)
36 Exercise - Type 2 Convergence · Level 2
\( \displaystyle\int_{0}^{4} \dfrac{d x}{x^2 - x - 2} \)
37 Exercise - Type 2 Convergence · Level 2
\( \displaystyle\int_{0}^{1} r \ln r d r \)
38 Exercise - Type 2 Convergence · Level 2
\( \displaystyle\int_{0}^{\dfrac{\pi}{2}} \dfrac{\cos \theta}{\sqrt{\sin \theta}} d \theta \)
39 Exercise - Type 2 Convergence · Level 3
\( \displaystyle\int_{-1}^0 \dfrac{e^{\dfrac{1}{x}}}{x^3} d x \)
40 Exercise - Type 2 Convergence · Level 3
\( \displaystyle\int_{0}^{1} \dfrac{e^{\dfrac{1}{x}}}{x^3} d x \)
41 Exercise - Area of Unbounded Region · Level 2
Sketch the region and find its area (if the area is finite). \(S = \{(x, y) | x \geq 1, 0 \leq y \leq e^{-x}\}\)
42 Exercise - Area of Unbounded Region · Level 2
Sketch the region and find its area (if the area is finite). \(S = \{(x, y) | x \leq 0, 0 \leq y \leq e^x\}\)
43 Exercise - Area of Unbounded Region · Level 2
Sketch the region and find its area (if the area is finite). \(S = \{(x, y) | x \geq 1, 0 \leq y \leq 1/(x^3 + x)\}\)
44 Exercise - Area of Unbounded Region · Level 2
Sketch the region and find its area (if the area is finite). \(S = \{(x, y) | x \geq 0, 0 \leq y \leq x e^{-x}\}\)
45 Exercise - Area of Unbounded Region · Level 2
Sketch the region and find its area (if the area is finite). \(S = \{(x, y) | 0 \leq x < \dfrac{\pi}{2}, 0 \leq y \leq \sec^2 x\}\)
46 Exercise - Area of Unbounded Region · Level 2
Sketch the region and find its area (if the area is finite). \(S = \{(x, y) | -2 < x \leq 0, 0 \leq y \leq \dfrac{1}{\sqrt{x+2}}\}\)
47 Exercise - Comparison Theorem Investigation · Level 3
(a) If \(g(x) = (\sin^2 x)/x^2\), use your calculator or computer to make a table of approximate values of \(\displaystyle\int_{1}^{t} g(x) d x\) for \(t = 2, 5, 10, 100, 1000\), and \(10000\). Does it appear that \(\displaystyle\int_{1}^{\infty} g(x) d x\) is convergent?
(b) Use the Comparison Theorem with \(f(x) = 1/x^2\) to show that \(\displaystyle\int_{1}^{\infty} g(x) d x\) is convergent.
(c) Illustrate part (b) by graphing \(f\) and \(g\) on the same screen for \(1 \leq x \leq 10\). Use your graph to explain intuitively why \(\displaystyle\int_{1}^{\infty} g(x) d x\) is convergent.

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48 Exercise - Comparison Theorem Investigation · Level 3
(a) If \(g(x) = 1/(\sqrt{x} - 1)\), use your calculator or computer to make a table of approximate values of \(\displaystyle\int_{2}^{t} g(x) d x\) for \(t = 5, 10, 100, 1000\), and \(10000\). Does it appear that \(\displaystyle\int_{2}^{\infty} g(x) d x\) is convergent or divergent?
(b) Use the Comparison Theorem with \(f(x) = \dfrac{1}{\sqrt{x}}\) to show that \(\displaystyle\int_{2}^{\infty} g(x) d x\) is divergent.
(c) Illustrate part (b) by graphing \(f\) and \(g\) on the same screen for \(2 \leq x \leq 20\). Use your graph to explain intuitively why \(\displaystyle\int_{2}^{\infty} g(x) d x\) is divergent.

Enter your answer directly below each part above.

49 Exercise - Comparison Test · Level 2
Use the Comparison Theorem to determine whether the integral is convergent or divergent. \(\displaystyle\int_{0}^{\infty} \dfrac{x}{x^3 + 1} d x\)
50 Exercise - Comparison Test · Level 2
Use the Comparison Theorem to determine whether the integral is convergent or divergent. \(\displaystyle\int_{1}^{\infty} \dfrac{1 + \sin^2 x}{\sqrt{x}} d x\)
51 Exercise - Comparison Test · Level 2
Use the Comparison Theorem to determine whether the integral is convergent or divergent. \(\displaystyle\int_{1}^{\infty} \dfrac{x + 1}{\sqrt{x^4 - x}} d x\)
52 Exercise - Comparison Test · Level 2
Use the Comparison Theorem to determine whether the integral is convergent or divergent. \(\displaystyle\int_{0}^{\infty} \dfrac{\arctan x}{2 + e^x} d x\)
53 Exercise - Comparison Test · Level 2
Use the Comparison Theorem to determine whether the integral is convergent or divergent. \(\displaystyle\int_{0}^{1} \dfrac{\sec^2 x}{x \sqrt{x}} d x\)
54 Exercise - Comparison Test · Level 2
Use the Comparison Theorem to determine whether the integral is convergent or divergent. \(\displaystyle\int_{0}^{\pi} \dfrac{\sin^2 x}{\sqrt{x}} d x\)
55 Exercise - Combined Type 1 & Type 2 · Level 3
The integral \(\displaystyle\int_{0}^{\infty} \dfrac{1}{\sqrt{x}(1+x)} d x\) is improper for two reasons: The interval \([0, \infty)\) is infinite and the integrand has an infinite discontinuity at \(0\). Evaluate it by expressing it as a sum of improper integrals of Type 2 and Type 1 as follows: \(\displaystyle\int_{0}^{\infty} \dfrac{1}{\sqrt{x}(1+x)} d x = \displaystyle\int_{0}^{1} \dfrac{1}{\sqrt{x}(1+x)} d x + \displaystyle\int_{1}^{\infty} \dfrac{1}{\sqrt{x}(1+x)} d x\)
56 Exercise - Combined Improper · Level 3
Evaluate \(\displaystyle\int_{2}^{\infty} \dfrac{1}{x \sqrt{x^2 - 4}} d x\) by the same method as in Exercise 55.
57 Exercise - p-Integral · Level 2
Find the values of \(p\) for which the integral converges and evaluate the integral for those values of \(p\). \(\displaystyle\int_{0}^{1} \dfrac{1}{x^p} d x\)
58 Exercise - p-Integral · Level 2
Find the values of \(p\) for which the integral converges and evaluate the integral for those values of \(p\). \(\displaystyle\int_{e}^{\infty} \dfrac{1}{x (\ln x)^p} d x\)
59 Exercise - p-Integral · Level 3
Find the values of \(p\) for which the integral converges and evaluate the integral for those values of \(p\). \(\displaystyle\int_{0}^{1} x^p \ln x d x\)
60 Exercise - Gamma Function · Level 3
(a) Evaluate the integral \(\displaystyle\int_{0}^{\infty} x^n e^{-x} d x\) for \(n = 0, 1, 2\), and \(3\).
(b) Guess the value of \(\displaystyle\int_{0}^{\infty} x^n e^{-x} d x\) when \(n\) is an arbitrary positive integer.
(c) Prove your guess using mathematical induction.

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61 Exercise - Cauchy Principal Value · Level 3
(a) Show that \(\displaystyle\int_{-\infty}^\infty x d x\) is divergent.
(b) Show that \(\operatorname*{lim}\limits_{t \rightarrow \infty} \displaystyle\int_{-t}^t x d x = 0\). This shows that we can't define \(\displaystyle\int_{-\infty}^\infty f(x) d x = \operatorname*{lim}\limits_{t \rightarrow \infty} \displaystyle\int_{-t}^t f(x) d x\)

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62 Exercise - Application: Average Speed · Level 3
The average speed of molecules in an ideal gas is \(\bar{v} = \dfrac{4}{\sqrt{\pi}} \left(\dfrac{M}{2 R T}\right)^{\dfrac{3}{2}} \displaystyle\int_{0}^{\infty} v^3 e^{-M v^2/(2 R T)} d v\) where \(M\) is the molecular weight of the gas, \(R\) is the gas constant, \(T\) is the gas temperature, and \(v\) is the molecular speed. Show that \(\bar{v} = \sqrt{\dfrac{8 R T}{\pi M}}\)
63 Exercise - Volume of Revolution · Level 3
We know from Example 1 that the region \(cal(R) = \{(x, y) | x \geq 1, 0 \leq y \leq \dfrac{1}{x}\}\) has infinite area. Show that by rotating \(cal(R)\) about the \(x\)-axis we obtain a solid with finite volume.
64 Exercise - Application: Work · Level 3
Use the information and data in Exercise 6.4.33 to find the work required to propel a 1000-kg space vehicle out of the earth's gravitational field.
65 Exercise - Application: Escape Velocity · Level 3
Find the escape velocity \(v_0\) that is needed to propel a rocket of mass \(m\) out of the gravitational field of a planet with mass \(M\) and radius \(R\). Use Newton's Law of Gravitation (see Exercise 6.4.33) and the fact that the initial kinetic energy of \(\dfrac{1}{2} m v_0^2\) supplies the needed work.
66 Exercise - Application: Stellar Stereography · Level 3
Astronomers use a technique called stellar stereography to determine the density of stars in a star cluster from the observed (two-dimensional) density that can be analyzed from a photograph. Suppose that in a spherical cluster of radius \(R\) the density of stars depends only on the distance \(r\) from the center of the cluster. If the perceived star density is given by \(y(s)\), where \(s\) is the observed planar distance from the center of the cluster, and \(x(r)\) is the actual density, it can be shown that \(y(s) = \displaystyle\int_{s}^{R} \dfrac{2 r}{\sqrt{r^2 - s^2}} x(r) d r\) If the actual density of stars in a cluster is \(x(r) = \dfrac{1}{2}(R - r)^2\), find the perceived density \(y(s)\).
67 Exercise - Application: Reliability · Level 2
A manufacturer of lightbulbs wants to produce bulbs that last about 700 hours but, of course, some bulbs burn out faster than others. Let \(F(t)\) be the fraction of the company's bulbs that burn out before \(t\) hours, so \(F(t)\) always lies between \(0\) and \(1\).
(a) Make a rough sketch of what you think the graph of \(F\) might look like.
(b) What is the meaning of the derivative \(r(t) = F'(t)\)?
(c) What is the value of \(\displaystyle\int_{0}^{\infty} r(t) d t\)? Why?

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68 Exercise - Application: Mean Life of Carbon-14 · Level 3
As we saw in Section 3.8, a radioactive substance decays exponentially: The mass at time \(t\) is \(m(t) = m(0) e^{k t}\), where \(m(0)\) is the initial mass and \(k\) is a negative constant. The mean life \(M\) of an atom in the substance is \(M = -k \displaystyle\int_{0}^{\infty} t e^{k t} d t\) For the radioactive carbon isotope, \({}^{14} C\), used in radiocarbon dating, the value of \(k\) is \(-0.000121\). Find the mean life of a \({}^{14} C\) atom.
69 Exercise - Application: Drug Use Spread · Level 3
In a study of the spread of illicit drug use from an enthusiastic user to a population of \(N\) users, the authors model the number of expected new users by the equation \(\gamma = \displaystyle\int_{0}^{\infty} \dfrac{c N (1 - e^{-k t})}{k} e^{-\lambda t} d t\) where \(c, k\) and \(\lambda\) are positive constants. Evaluate this integral to express \(\gamma\) in terms of \(c, N, k\), and \(\lambda\).
70 Exercise - Application: Dialysis · Level 3
Dialysis treatment removes urea and other waste products from a patient's blood by diverting some of the bloodflow externally through a machine called a dialyzer. The rate at which urea is removed from the blood (in mg/min) is often well described by the equation \(u(t) = \dfrac{r}{V} C_0 e^{-r \dfrac{t}{V}}\) where \(r\) is the rate of flow of blood through the dialyzer (in mL/min), \(V\) is the volume of the patient's blood (in mL), and \(C_0\) is the amount of urea in the blood (in mg) at time \(t = 0\). Evaluate the integral \(\displaystyle\int_{0}^{\infty} u(t) d t\) and interpret it.
71 Exercise - Tail Estimate · Level 2
Determine how large the number \(a\) has to be so that \(\displaystyle\int_{a}^{\infty} \dfrac{1}{x^2 + 1} d x < 0.001\)
72 Exercise - Numerical Estimation · Level 3
Estimate the numerical value of \(\displaystyle\int_{0}^{\infty} e^{-x^2} d x\) by writing it as the sum of \(\displaystyle\int_{0}^{4} e^{-x^2} d x\) and \(\displaystyle\int_{4}^{\infty} e^{-x^2} d x\). Approximate the first integral by using Simpson's Rule with \(n = 8\) and show that the second integral is smaller than \(\displaystyle\int_{4}^{\infty} e^{-4 x} d x\), which is less than \(0.0000001\).
73 Exercise - Laplace Transform · Level 3
If \(f(t)\) is continuous for \(t \geq 0\), the Laplace transform of \(f\) is the function \(F\) defined by \(F(s) = \displaystyle\int_{0}^{\infty} f(t) e^{-s t} d t\) and the domain of \(F\) is the set consisting of all numbers \(s\) for which the integral converges. Find the Laplace transforms of the following functions.
(a) \(f(t) = 1\)
(b) \(f(t) = e^t\)
(c) \(f(t) = t\)

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74 Exercise - Laplace Transform Existence · Level 3
Show that if \(0 \leq f(t) \leq M e^{a t}\) for \(t \geq 0\), where \(M\) and \(a\) are constants, then the Laplace transform \(F(s)\) exists for \(s > a\).
75 Exercise - Laplace Transform Property · Level 3
Suppose that \(0 \leq f(t) \leq M e^{a t}\) and \(0 \leq f'(t) \leq K e^{a t}\) for \(t \geq 0\), where \(f'\) is continuous. If the Laplace transform of \(f(t)\) is \(F(s)\) and the Laplace transform of \(f'(t)\) is \(G(s)\), show that \(G(s) = s F(s) - f(0) \quad s > a\)
76 Exercise - Independence of Splitting Point · Level 3
If \(\displaystyle\int_{-\infty}^\infty f(x) d x\) is convergent and \(a\) and \(b\) are real numbers, show that \(\displaystyle\int_{-\infty}^a f(x) d x + \displaystyle\int_{a}^{\infty} f(x) d x = \displaystyle\int_{-\infty}^b f(x) d x + \displaystyle\int_{b}^{\infty} f(x) d x\)
77 Exercise - Integration by Parts · Level 3
Show that \(\displaystyle\int_{0}^{\infty} x^2 e^{-x^2} d x = \dfrac{1}{2} \displaystyle\int_{0}^{\infty} e^{-x^2} d x\).
78 Exercise - Area Interpretation · Level 3
Show that \(\displaystyle\int_{0}^{\infty} e^{-x^2} d x = \displaystyle\int_{0}^{1} \sqrt{-\ln y} d y\) by interpreting the integrals as areas.
79 Exercise - Convergent Combination · Level 3
Find the value of the constant \(C\) for which the integral \(\displaystyle\int_{0}^{\infty} \left(\dfrac{1}{\sqrt{x^2 + 4}} - \dfrac{C}{x + 2}\right) d x\) converges. Evaluate the integral for this value of \(C\).
80 Exercise - Convergent Combination · Level 3
Find the value of the constant \(C\) for which the integral \(\displaystyle\int_{0}^{\infty} \left(\dfrac{x}{x^2 + 1} - \dfrac{C}{3 x + 1}\right) d x\) converges. Evaluate the integral for this value of \(C\).
81 Exercise - Conceptual · Level 2
Suppose \(f\) is continuous on \([0, \infty)\) and \(\operatorname*{lim}\limits_{x \rightarrow \infty} f(x) = 1\). Is it possible that \(\displaystyle\int_{0}^{\infty} f(x) d x\) is convergent?
82 Exercise - Generalized p-Integral · Level 3
Show that if \(a > -1\) and \(b > a + 1\), then the following integral is convergent. \(\displaystyle\int_{0}^{\infty} \dfrac{x^a}{1 + x^b} d x\)
83 Example - Type 1 Improper Integral · Level 2
Determine whether the integral \(\displaystyle\int_{1}^{\infty} \dfrac{1}{x} d x\) is convergent or divergent.
84 Example - Type 1 Improper Integral · Level 3
Evaluate \(\displaystyle\int_{-\infty}^0 x e^x d x\).
85 Example - Two-sided Improper Integral · Level 3
Evaluate \(\displaystyle\int_{-\infty}^\infty \dfrac{1}{1+x^2} d x\).
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86 Example - p-Integral · Level 3
For what values of \(p\) is the integral \(\displaystyle\int_{1}^{\infty} \dfrac{1}{x^p} d x\) convergent?
87 Example - Type 2 Improper Integral · Level 2
Find \(\displaystyle\int_{2}^{5} \dfrac{1}{\sqrt{x-2}} d x\).
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88 Example - Type 2 Improper Integral · Level 2
Determine whether \(\displaystyle\int_{0}^{\dfrac{\pi}{2}} \sec x d x\) converges or diverges.
89 Example - Type 2 Improper Integral with Interior Discontinuity · Level 2
Evaluate \(\displaystyle\int_{0}^{3} \dfrac{d x}{x-1}\) if possible.
90 Example - Type 2 Improper Integral · Level 3
Evaluate \(\displaystyle\int_{0}^{1} \ln x d x\).
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91 Example - Comparison Test · Level 3
Show that \(\displaystyle\int_{0}^{\infty} e^{-x^2} d x\) is convergent.
92 Example - Comparison Test (Divergence) · Level 2
Show that \(\displaystyle\int_{1}^{\infty} \dfrac{1+e^{-x}}{x} d x\) is divergent.

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