Chapter 7, Techniques of Integration Video Solutions, Calculus

Section 8

Improper Integrals

Problem 1

Explain why each of the following integrals is improper.

(a) $ \displaystyle \int_1^2 \frac{x}{x - 1}\ dx $
(b) $ \displaystyle \int_0^\infty \frac{1}{1 + x^3}\ dx $
(c) $ \displaystyle \int_{-\infty}^\infty x^2 e^{-x^2}\ dx $
(d) $ \displaystyle \int_0^{\frac{\pi}{4}} \cot x\ dx $

Wen Z.

Numerade Educator

01:32

Problem 2

Which of the following integrals are improper? Why?

(a) $ \displaystyle \int_0^{\frac{\pi}{4}} \tan x\ dx $
(b) $ \displaystyle \int_0^\pi \tan x\ dx $
(c) $ \displaystyle \int_{-1}^1 \frac{dx}{x^2 - x - 2} $
(d) $ \displaystyle \int_0^\infty e^{-x^3}\ dx $

Wen Z.

Numerade Educator

09:31

Problem 3

Find the area under the curve $ y = \frac{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 \ge 1 $.

Willis J.

Numerade Educator

03:08

Problem 4

(a) Graph the functions $ f(x) = \frac{1}{x^{1.1}} $ and $ g(x) = \frac{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} $, and $ 10^{20} $.
(c) Find the total area under each curve for $ x \ge 1 $, if it exists.

Wen Z.

Numerade Educator

02:21

Problem 5

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_3^\infty \frac{1}{(x - 2)^{\frac{3}{2}}}\ dx $

Wen Z.

Numerade Educator

03:38

Problem 6

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_0^\infty \frac{1}{\sqrt[4]{1 + x}}\ dx $

Wen Z.

Numerade Educator

02:03

Problem 7

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_{-\infty}^0 \frac{1}{3 - 4x}\ dx $

Wen Z.

Numerade Educator

03:43

Problem 8

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_1^\infty \frac{1}{(2x + 1)^3}\ dx $

Wen Z.

Numerade Educator

03:24

Problem 9

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_2^\infty e^{-5p}\ dp $

Wen Z.

Numerade Educator

03:29

Problem 10

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_{-\infty}^0 2^r\ dr $

Wen Z.

Numerade Educator

04:47

Problem 11

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_0^\infty \frac{x^2}{\sqrt{1 + x^3}}\ dx $

Wen Z.

Numerade Educator

03:13

Problem 12

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_{-\infty}^\infty (y^3 - 3y^2)\ dy $

Wen Z.

Numerade Educator

02:54

Problem 13

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_{-\infty}^\infty xe^{-x^2}\ dx $

Wen Z.

Numerade Educator

02:17

Problem 14

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_1^\infty \frac{e^{\frac{-1}{x}}}{x^2}\ dx $

Wen Z.

Numerade Educator

03:00

Problem 15

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_0^\infty \sin^2 \alpha\ d \alpha $

Wen Z.

Numerade Educator

02:32

Problem 16

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_0^\infty \sin \theta e^{\cos \theta}\ d \theta $

Wen Z.

Numerade Educator

03:38

Problem 17

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_1^\infty \frac{1}{x^2 + x}\ dx $

Wen Z.

Numerade Educator

03:13

Problem 18

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_2^\infty \frac{dv}{v^2 + 2v - 3} $

Wen Z.

Numerade Educator

03:25

Problem 19

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_{-\infty}^0 ze^{2z}\ dz $

Wen Z.

Numerade Educator

04:25

Problem 20

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_2^\infty ye^{-3y}\ dy $

Wen Z.

Numerade Educator

02:22

Problem 21

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_1^\infty \frac{\ln x}{x}\ dx $

Wen Z.

Numerade Educator

03:21

Problem 22

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_1^\infty \frac{\ln x}{x^2}\ dx $

Wen Z.

Numerade Educator

03:30

Problem 23

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_{-\infty}^0 \frac{z}{z^4 + 4}\ dz $

Wen Z.

Numerade Educator

02:26

Problem 24

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_e^\infty \frac{1}{x (\ln x)^2}\ dx $

Wen Z.

Numerade Educator

04:01

Problem 25

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_0^\infty e^{-\sqrt{y}}\ dy $

Wen Z.

Numerade Educator

03:41

Problem 26

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_1^\infty \frac{dx}{\sqrt{x} + x \sqrt{x}} $

Wen Z.

Numerade Educator

01:47

Problem 27

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_0^1 \frac{1}{x}\ dx $

Wen Z.

Numerade Educator

04:41

Problem 28

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_0^5 \frac{1}{\sqrt[3]{5 - x}}\ dx $

Wen Z.

Numerade Educator

03:19

Problem 29

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_{-2}^{14} \frac{dx}{\sqrt[4]{x + 2}} $

Wen Z.

Numerade Educator

03:56

Problem 30

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_{-1}^2 \frac{x}{(x + 1)^2}\ dx $

Wen Z.

Numerade Educator

03:18

Problem 31

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_{-2}^3 \frac{1}{x^4}\ dx $

Wen Z.

Numerade Educator

01:37

Problem 32

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_0^1 \frac{dx}{\sqrt{1 - x^2}} $

Wen Z.

Numerade Educator

03:23

Problem 33

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_0^9 \frac{1}{\sqrt[3]{x - 1}}\ dx $

Wen Z.

Numerade Educator

02:45

Problem 34

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_0^5 \frac{w}{w - 2}\ dw $

Wen Z.

Numerade Educator

02:10

Problem 35

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_0^{\frac{\pi}{2}} \tan^2 \theta\ d \theta $

Wen Z.

Numerade Educator

03:57

Problem 36

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_0^4 \frac{dx}{x^2 - x - 2} $

Wen Z.

Numerade Educator

03:58

Problem 37

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_0^1 r \ln r\ dr $

Wen Z.

Numerade Educator

03:20

Problem 38

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_0^{\frac{\pi}{2}} \frac{\cos \theta}{\sqrt{\sin \theta}}\ d \theta $

Wen Z.

Numerade Educator

04:27

Problem 39

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_{-1}^0 \frac{e^{\frac{1}{x}}}{x^3}\ dx $

Wen Z.

Numerade Educator

03:44

Problem 40

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

$ \displaystyle \int_0^1 \frac{e^{\frac{1}{x}}}{x^3}\ dx $

Wen Z.

Numerade Educator

01:56

Problem 41

Sketch the region and find its area (if the area is finite).

$ S = \{ (x, y) \mid x \ge 1, 0 \le y \le e^{-x} \} $

Wen Z.

Numerade Educator

01:40

Problem 42

Sketch the region and find its area (if the area is finite).

$ S = \{ (x, y) \mid x \le 0, 0 \le y \le e^x \} $

Wen Z.

Numerade Educator

06:53

Problem 43

Sketch the region and find its area (if the area is finite).

$ S = \{ (x, y) \mid x \ge 1, 0 \le y \le \frac{1}{(x^3 + x)} \} $

Wen Z.

Numerade Educator

04:06

Problem 44

Sketch the region and find its area (if the area is finite).

$ S = \{ (x, y) \mid x \ge 0, 0 \le y \le xe^{-x} \} $

Wen Z.

Numerade Educator

02:06

Problem 45

Sketch the region and find its area (if the area is finite).

$ S = \{ (x, y) \mid 0 \le x < \frac{\pi}{2}, 0 \le y \le \sec^2 x \} $

Wen Z.

Numerade Educator

03:13

Problem 46

Sketch the region and find its area (if the area is finite).

$ S = \{ (x, y) \mid -2 < x \le 0, 0 \le y \le \frac{1}{\sqrt{x + 2}} $

Wen Z.

Numerade Educator

04:26

Problem 47

(a) If $ g(x) = \frac{(\sin^2 x)}{x^2} $, use your calculator or computer to make a table of approximate values of $ \displaystyle \int_1^t g(x)\ dx $ for $ t = 2, 5, 10, 100, 1000 $, and $ 10,000 $. Does it appear that $ \displaystyle \int_1^\infty g(x)\ dx $ is convergent?
(b) Use the Comparison Theorem with $ f(x) = \frac{1}{x^2} $ to show that $ \displaystyle \int_1^\infty g(x)\ dx $ is convergent.
(c) Illustrate part (b) by graphing $ f $ and $ g $ on the same screen for $ 1 \le x \le 10 $. Use your graph to explain intuitively why $ \displaystyle \int_1^\infty g(x)\ dx $ is convergent.

Wen Z.

Numerade Educator

03:50

Problem 48

(a) If $ g(x) = \frac{1}{(\sqrt{x} - 1)} $, use your calculator or computer to make a table of approximate values of $ \displaystyle \int_2^t g(x)\ dx $ for $ t = 5, 10, 100, 1000 $, and $ 10,000 $. Does it appear that $ \displaystyle \int_2^\infty g(x)\ dx $ is convergent or divergent?
(b) Use the Comparison Theorem with $ f(x) = \frac{1}{\sqrt{x}} $ to show that $ \displaystyle \int_2^\infty g(x)\ dx $ is divergent.
(c) Illustrate part (b) by graphing $ f $ and $ g $ on the same screen for $ 2 \le x \le 20 $. Use your graph to explain intuitively why $ \displaystyle \int_2^\infty g(x)\ dx $ is divergent.

Wen Z.

Numerade Educator

02:00

Problem 49

Use the Comparison Theorem to determine whether the integral is convergent or divergent.

$ \displaystyle \int_0^\infty \frac{x}{x^3 + 1}\ dx $

Willis J.

Numerade Educator

02:27

Problem 50

Use the Comparison Theorem to determine whether the integral is convergent or divergent.

$ \displaystyle \int_1^\infty \frac{1 + \sin^2 x}{\sqrt{x}}\ dx $

Wen Z.

Numerade Educator

02:16

Problem 51

Use the Comparison Theorem to determine whether the integral is convergent or divergent.

$ \displaystyle \int_1^\infty \frac{x + 1}{\sqrt{x^4 - x}}\ dx $

Willis J.

Numerade Educator

02:49

Problem 52

Use the Comparison Theorem to determine whether the integral is convergent or divergent.

$ \displaystyle \int_0^\infty \frac{\arctan x}{2 + e^x}\ dx $

Wen Z.

Numerade Educator

02:16

Problem 53

Use the Comparison Theorem to determine whether the integral is convergent or divergent.

$ \displaystyle \int_0^1 \frac{\sec^2 x}{x \sqrt{x}}\ dx $

Wen Z.

Numerade Educator

01:45

Problem 54

Use the Comparison Theorem to determine whether the integral is convergent or divergent.

$ \displaystyle \int_0^\pi \frac{\sin^2 x}{\sqrt{x}}\ dx $

Wen Z.

Numerade Educator

04:49

Problem 55

The integral
$$ \int_0^\infty \frac{1}{\sqrt{x} (1 + x)}\ dx $$
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:
$$ \int_0^\infty \frac{1}{\sqrt{x} (1 + x)}\ dx = \int_0^1 \frac{1}{\sqrt{x} (1 + x)}\ dx + \int_1^\infty \frac{1}{\sqrt{x} (1 + x)}\ dx $$

Wen Z.

Numerade Educator

12:57

Problem 56

Evaluate
$$ \displaystyle \int_2^\infty \frac{1}{x \sqrt{x^2 - 4}}\ dx $$
by the same method as in Exercise 55.

Chris T.

Numerade Educator

05:17

Problem 57

Find the values of $ p $ for which the integral converges and evaluate the integral for those values of $ p $.

$ \displaystyle \int_0^1 \frac{1}{x^p}\ dx $

Willis J.

Numerade Educator

07:03

Problem 58

Find the values of $ p $ for which the integral converges and evaluate the integral for those values of $ p $.

$ \displaystyle \int_e^\infty \frac{1}{x (\ln x)^p}\ dx $

Willis J.

Numerade Educator

08:57

Problem 59

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\ dx $

Wen Z.

Numerade Educator